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Relations among eta functions of level 8

Related to http://www.risc.jku.at/people/hemmecke/eta/

fricas
-- eta relations --
g1 := E1^8*E2^6*E4^10-E1^12*E2^8*E8^4-4*E1^4*E2^8*E4^8*E8^4+32*E2^10*E4^6*E8^8-16*E1^12*E8^12-256*E1^4*E4^8*E8^12;
Type: Polynomial(Integer)
fricas
g2 := E2^10*E4^8-E1^12*E4^2*E8^4-8*E1^4*E4^10*E8^4-4*E1^8*E2^2*E8^8;
Type: Polynomial(Integer)
fricas
g3 := E2^8*E4^10-E1^4*E2^10*E8^4-4*E1^8*E4^2*E8^8-32*E1^4*E2^2*E8^12;
Type: Polynomial(Integer)
fricas
g4 := E2^12-E1^8*E4^4-8*E1^4*E2^2*E4^2*E8^4;
Type: Polynomial(Integer)
fricas
g5 := E4^12-E1^4*E2^2*E4^2*E8^4-4*E2^4*E8^8;
Type: Polynomial(Integer)
fricas
-- Somos eta relations --
q8_12_24 := -E2^12+E1^8*E4^4+8*E4^12-32*E2^4*E8^8;
Type: Polynomial(Integer)
fricas
t8_12_24 := -E2^12+E1^8*E4^4+8*E1^4*E2^2*E4^2*E8^4;
Type: Polynomial(Integer)
fricas
t8_12_48 := -E4^12+E1^4*E2^2*E4^2*E8^4+4*E2^4*E8^8;
Type: Polynomial(Integer)
fricas
q8_18_36 := -E1^4*E2^12*E4^2+E1^12*E4^6+16*E1^4*E4^14-8*E2^14*E8^4;
Type: Polynomial(Integer)
fricas
x8_18_36a := -E1^4*E2^12*E4^2+E1^12*E4^6+8*E1^4*E4^14-32*E2^2*E4^12*E8^4+128*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
x8_18_36b := -E1^4*E2^12*E4^2+E1^12*E4^6+8*E2^14*E8^4-64*E2^2*E4^12*E8^4+256*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
q8_18_52 := -E2^10*E4^8+E1^12*E4^2*E8^4+8*E1^4*E4^10*E8^4+4*E1^8*E2^2*E8^8;
Type: Polynomial(Integer)
fricas
q8_18_56 := -E2^8*E4^10+E1^4*E2^10*E8^4+4*E1^8*E4^2*E8^8+32*E1^4*E2^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_18_60 := E1^4*E4^14-E2^14*E8^4+4*E2^2*E4^12*E8^4-16*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
t8_18_60a := E1^4*E4^14-E2^14*E8^4+4*E1^4*E2^4*E4^2*E8^8;
Type: Polynomial(Integer)
fricas
t8_18_60b := -2*E1^4*E4^14+E2^14*E8^4+E1^8*E2^2*E4^4*E8^4;
Type: Polynomial(Integer)
fricas
x8_20_40 := -E2^20+E1^8*E2^8*E4^4+8*E2^8*E4^12-32*E1^8*E4^4*E8^8-256*E1^4*E2^2*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
x8_20_56 := -E2^12*E4^8+8*E4^20+E1^12*E2^2*E4^2*E8^4+4*E1^8*E2^4*E8^8-32*E2^4*E4^8*E8^8;
Type: Polynomial(Integer)
fricas
q8_20_60a := -E2^10*E4^10+E1^4*E2^12*E8^4+4*E1^4*E4^12*E8^4+16*E1^4*E2^4*E8^12;
Type: Polynomial(Integer)
fricas
q8_20_60b := -E2^10*E4^10+E1^12*E4^4*E8^4+12*E1^4*E4^12*E8^4-16*E1^4*E2^4*E8^12;
Type: Polynomial(Integer)
fricas
q8_20_60c := -2*E2^10*E4^10+E1^4*E2^12*E8^4+E1^12*E4^4*E8^4+16*E1^4*E4^12*E8^4;
Type: Polynomial(Integer)
fricas
q8_20_60d := 2*E2^10*E4^10-3*E1^4*E2^12*E8^4+E1^12*E4^4*E8^4-64*E1^4*E2^4*E8^12;
Type: Polynomial(Integer)
fricas
q8_20_60e := -E2^10*E4^10+E1^4*E2^12*E8^4+8*E1^4*E4^12*E8^4-4*E1^8*E2^2*E4^2*E8^8;
Type: Polynomial(Integer)
fricas
q8_20_60f := -E2^10*E4^10+E1^12*E4^4*E8^4+12*E1^8*E2^2*E4^2*E8^8+32*E1^4*E2^4*E8^12;
Type: Polynomial(Integer)
fricas
q8_22_64a := -E2^12*E4^10+E1^8*E4^14+8*E1^8*E2^4*E4^2*E8^8+32*E1^4*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
q8_22_64b := E2^12*E4^10+E1^8*E4^14-2*E1^4*E2^14*E8^4-32*E1^4*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
q8_22_64c := E2^12*E4^10-3*E1^8*E4^14+2*E1^12*E2^2*E4^4*E8^4-32*E1^4*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
q8_22_68a := -E2^10*E4^12+E1^12*E4^6*E8^4+4*E2^14*E8^8+8*E1^8*E2^2*E4^4*E8^8;
Type: Polynomial(Integer)
fricas
q8_22_68b := -E2^10*E4^12+E1^12*E4^6*E8^4+12*E2^14*E8^8-64*E1^4*E2^4*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_22_68c := -E2^10*E4^12+E1^12*E4^6*E8^4+16*E1^4*E4^14*E8^4-4*E2^14*E8^8;
Type: Polynomial(Integer)
fricas
x8_24_48 := -E2^24+E1^8*E2^12*E4^4+8*E1^8*E4^16+32*E2^16*E8^8-256*E1^4*E2^6*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_24_48a := -E2^24+E1^8*E2^12*E4^4+16*E1^8*E4^16-8*E1^12*E2^2*E4^6*E8^4;
Type: Polynomial(Integer)
fricas
q8_24_48b := -E2^24+E1^8*E2^12*E4^4+8*E1^8*E4^16+32*E1^8*E2^4*E4^4*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_48c := -E2^24+E1^16*E4^8+16*E2^12*E4^12-128*E1^4*E2^2*E4^14*E8^4;
Type: Polynomial(Integer)
fricas
x8_24_72 := -E1^8*E4^16+8*E4^24+E1^12*E2^2*E4^6*E8^4-4*E2^16*E8^8-32*E2^4*E4^12*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_72a := -E2^12*E4^12+E1^8*E4^16+8*E2^16*E8^8-32*E1^4*E2^6*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_24_72b := E2^12*E4^12-2*E1^8*E4^16+E1^12*E2^2*E4^6*E8^4-4*E2^16*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_72c := -E1^8*E4^16+E1^12*E2^2*E4^6*E8^4+8*E1^4*E2^2*E4^14*E8^4-4*E2^16*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_72d := -E1^8*E4^16+E1^4*E2^14*E4^2*E8^4-4*E2^16*E8^8+32*E1^4*E2^6*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_24_72e := -E1^8*E4^16+E1^12*E2^2*E4^6*E8^4+4*E2^16*E8^8-32*E1^4*E2^6*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_24_72f := -E2^12*E4^12+E1^8*E4^16+4*E2^16*E8^8+4*E1^8*E2^4*E4^4*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_72g := -E1^8*E4^16+E1^4*E2^14*E4^2*E8^4-8*E1^4*E2^2*E4^14*E8^4+4*E2^16*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_72h := -E2^12*E4^12+4*E4^24+E1^4*E2^14*E4^2*E8^4-64*E2^8*E8^16;
Type: Polynomial(Integer)
fricas
q8_24_96a := -2*E4^24+E2^16*E8^8+E1^8*E2^4*E4^4*E8^8+8*E2^4*E4^12*E8^8;
Type: Polynomial(Integer)
fricas
q8_24_96b := -E4^24+E2^16*E8^8+4*E2^4*E4^12*E8^8-4*E1^4*E2^6*E4^2*E8^12;
Type: Polynomial(Integer)
fricas
q8_26_76 := E1^4*E2^8*E4^14-E2^22*E8^4+4*E1^12*E4^6*E8^8+32*E1^8*E2^2*E4^4*E8^12;
Type: Polynomial(Integer)
fricas
q8_26_92 := -2*E1^4*E4^22+E2^14*E4^8*E8^4+E1^12*E2^4*E4^2*E8^8+4*E1^8*E2^6*E8^12;
Type: Polynomial(Integer)
fricas
-- Relation relations --
relationRelations := [_ q8_12_24+g4-8*g5,_ t8_12_24+g4,_ t8_12_48+g5,_ q8_18_36+(E1^4*E4^2+8*E2^2*E8^4)*g4-16*E1^4*E4^2*g5,_ x8_18_36a+E1^4*E4^2*g4+(-8*E1^4*E4^2+32*E2^2*E8^4)*g5,_ x8_18_36b+(E1^4*E4^2-8*E2^2*E8^4)*g4+64*E2^2*E8^4*g5,_ q8_18_52+g2,_ q8_18_56+g3,_ q8_18_60+E2^2*E8^4*g4+(-E1^4*E4^2-4*E2^2*E8^4)*g5,_ t8_18_60a+E2^2*E8^4*g4-E1^4*E4^2*g5,_ t8_18_60b-E2^2*E8^4*g4+2*E1^4*E4^2*g5,_ x8_20_40-8*E4^2*g3+E2^8*g4,_ x8_20_56+E2^2*g2-8*E4^8*g5,_ q8_20_60a+E4^2*g2-E1^4*E8^4*g4+4*E1^4*E8^4*g5,_ q8_20_60b+E4^2*g2-4*E1^4*E8^4*g5,_ q8_20_60c+2*E4^2*g2-E1^4*E8^4*g4,_ q8_20_60d-2*E4^2*g2+3*E1^4*E8^4*g4-16*E1^4*E8^4*g5,_ q8_20_60e+E4^2*g2-E1^4*E8^4*g4,_ q8_20_60f+E4^2*g2+8*E1^4*E8^4*g5,_ q8_22_64a+E2^2*E4^2*g2+(-E1^8*E4^2+8*E1^4*E2^2*E8^4)*g5,_ q8_22_64b-E2^2*E4^2*g2+2*E1^4*E2^2*E8^4*g4+(-E1^8*E4^2-8*E1^4*E2^2*E8^4)*g5,_ q8_22_64c-E2^2*E4^2*g2+(3*E1^8*E4^2-8*E1^4*E2^2*E8^4)*g5,_ q8_22_68a+E4^4*g2-4*E2^2*E8^8*g4+8*E1^4*E4^2*E8^4*g5,_ q8_22_68b+E4^4*g2-12*E2^2*E8^8*g4+8*E1^4*E4^2*E8^4*g5,_ q8_22_68c+E4^4*g2+4*E2^2*E8^8*g4-8*E1^4*E4^2*E8^4*g5,_ x8_24_48+(E2^12+8*E1^4*E2^2*E4^2*E8^4-32*E2^4*E8^8)*g4-8*E1^8*E4^4*g5,_ q8_24_48a+(E2^12+8*E1^4*E2^2*E4^2*E8^4)*g4-16*E1^8*E4^4*g5,_ q8_24_48b+(E2^12+8*E1^4*E2^2*E4^2*E8^4)*g4-8*E1^8*E4^4*g5,_ q8_24_48c-16*E2^2*E4^4*g2+(E2^12+E1^8*E4^4+8*E1^4*E2^2*E4^2*E8^4)*g4,_ x8_24_72+4*E2^4*E8^8*g4+(E1^8*E4^4-8*E4^12-8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72a+E2^2*E4^4*g2-8*E2^4*E8^8*g4+(-E1^8*E4^4+8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72b-E2^2*E4^4*g2+4*E2^4*E8^8*g4+(2*E1^8*E4^4-8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72c+4*E2^4*E8^8*g4+(E1^8*E4^4-8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72d+(-E1^4*E2^2*E4^2*E8^4+4*E2^4*E8^8)*g4+E1^8*E4^4*g5,_ q8_24_72e-4*E2^4*E8^8*g4+E1^8*E4^4*g5,_ q8_24_72f+E2^2*E4^4*g2-4*E2^4*E8^8*g4+(-E1^8*E4^4+8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72g+(-E1^4*E2^2*E4^2*E8^4-4*E2^4*E8^8)*g4+(E1^8*E4^4+8*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_72h+E2^2*E4^4*g2-E1^4*E2^2*E4^2*E8^4*g4+(-4*E4^12+4*E1^4*E2^2*E4^2*E8^4-16*E2^4*E8^8)*g5,_ q8_24_96a-E2^4*E8^8*g4+(2*E4^12+2*E1^4*E2^2*E4^2*E8^4)*g5,_ q8_24_96b-E2^4*E8^8*g4+(E4^12+E1^4*E2^2*E4^2*E8^4)*g5,_ q8_26_76-E1^4*E4^4*g3+(E2^10*E8^4+8*E1^4*E4^2*E8^8)*g4,_ q8_26_92-E2^4*E8^4*g2+(2*E1^4*E4^10+2*E1^8*E2^2*E8^4)*g5];
Type: List(Polynomial(Integer))
fricas
relationRelations

\label{eq1}\begin{array}{@{}l}
\displaystyle
\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \right.
\
\
\displaystyle
\left.\: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right] 
(1)
Type: List(Polynomial(Integer))




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