# Properties

 Label 15625.a.15625.1 Conductor $15625$ Discriminant $15625$ Mordell-Weil group $$\Z \oplus \Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{RM}$$ $$\End(J) \otimes \Q$$ $$\mathsf{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

This curve is isomorphic to the quotient of the modular curve $X_0(125)$ by its Fricke involution $w_{125}$; this quotient is also denoted $X_0^+(125)$.

## Simplified equation

 $y^2 + (x^3 + x + 1)y = 2x^5 + 2x^4 + 2x^3 + x^2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = 2x^5z + 2x^4z^2 + 2x^3z^3 + x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 + 8x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 2, 2, 2]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 2, 2, 2], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([1, 2, 5, 10, 10, 8, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$15625$$ $$=$$ $$5^{6}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$15625$$ $$=$$ $$5^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$I_4$$ $$=$$ $$25$$ $$=$$ $$5^{2}$$ $$I_6$$ $$=$$ $$1565$$ $$=$$ $$5 \cdot 313$$ $$I_{10}$$ $$=$$ $$640$$ $$=$$ $$2^{7} \cdot 5$$ $$J_2$$ $$=$$ $$25$$ $$=$$ $$5^{2}$$ $$J_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_6$$ $$=$$ $$-2500$$ $$=$$ $$- 2^{2} \cdot 5^{4}$$ $$J_8$$ $$=$$ $$-15625$$ $$=$$ $$- 5^{6}$$ $$J_{10}$$ $$=$$ $$15625$$ $$=$$ $$5^{6}$$ $$g_1$$ $$=$$ $$625$$ $$g_2$$ $$=$$ $$0$$ $$g_3$$ $$=$$ $$-100$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

Known points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : 1 : 2),\, (-1 : -4 : 2)$$
Known points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : 1 : 2),\, (-1 : -4 : 2)$$
Known points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : -5 : 2),\, (-1 : 5 : 2)$$

magma: [C![-1,-4,2],C![-1,1,2],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-1,-5,2],C![-1,5,2],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

## Mordell-Weil group of the Jacobian

Group structure: $$\Z \oplus \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.208772$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.234921$$ $$\infty$$
Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.208772$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.234921$$ $$\infty$$
Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - xz^2 - z^3$$ $$0.208772$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + xz^2 - z^3$$ $$0.234921$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.048361$$ Real period: $$13.02621$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.629964$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$6$$ $$6$$ $$1$$ $$1$$

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.12.2 no
$$3$$ 3.2592.2 no

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);