# Properties

 Label 2156.b.34496.1 Conductor $2156$ Discriminant $34496$ Mordell-Weil group $$\Z \oplus \Z/{6}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = 2x^5 - x^4 - 5x^3 + 3x + 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = 2x^5z - x^4z^2 - 5x^3z^3 + 3xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 8x^5 - 3x^4 - 18x^3 + x^2 + 12x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 0, -5, -1, 2]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 0, -5, -1, 2], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([4, 12, 1, -18, -3, 8]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$2156$$ $$=$$ $$2^{2} \cdot 7^{2} \cdot 11$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$34496$$ $$=$$ $$2^{6} \cdot 7^{2} \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2916$$ $$=$$ $$2^{2} \cdot 3^{6}$$ $$I_4$$ $$=$$ $$41745$$ $$=$$ $$3 \cdot 5 \cdot 11^{2} \cdot 23$$ $$I_6$$ $$=$$ $$37024569$$ $$=$$ $$3^{2} \cdot 563 \cdot 7307$$ $$I_{10}$$ $$=$$ $$4415488$$ $$=$$ $$2^{13} \cdot 7^{2} \cdot 11$$ $$J_2$$ $$=$$ $$729$$ $$=$$ $$3^{6}$$ $$J_4$$ $$=$$ $$20404$$ $$=$$ $$2^{2} \cdot 5101$$ $$J_6$$ $$=$$ $$734800$$ $$=$$ $$2^{4} \cdot 5^{2} \cdot 11 \cdot 167$$ $$J_8$$ $$=$$ $$29836496$$ $$=$$ $$2^{4} \cdot 17 \cdot 43 \cdot 2551$$ $$J_{10}$$ $$=$$ $$34496$$ $$=$$ $$2^{6} \cdot 7^{2} \cdot 11$$ $$g_1$$ $$=$$ $$205891132094649/34496$$ $$g_2$$ $$=$$ $$1976231914389/8624$$ $$g_3$$ $$=$$ $$2218766175/196$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points
$$(1 : 0 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : 0 : 2)$$
$$(1 : -2 : 1)$$ $$(-1 : 2 : 2)$$
All points
$$(1 : 0 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : 0 : 2)$$
$$(1 : -2 : 1)$$ $$(-1 : 2 : 2)$$
All points
$$(1 : 0 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -2 : 1)$$ $$(0 : 2 : 1)$$ $$(1 : -2 : 1)$$ $$(1 : 2 : 1)$$
$$(-1 : -2 : 2)$$ $$(-1 : 2 : 2)$$

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

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

Number of rational Weierstrass points: $$2$$

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: $$\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.123038$$ $$\infty$$
$$(-1 : 2 : 2) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (2x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-3xz^2 - z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.123038$$ $$\infty$$
$$(-1 : 2 : 2) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (2x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-3xz^2 - z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(0 : -2 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2 - 2z^3$$ $$0.123038$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (2x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^2z - 5xz^2 - 2z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.123038$$ Real period: $$21.05583$$ Tamagawa product: $$4$$ Torsion order: $$6$$ Leading coefficient: $$0.287853$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$6$$ $$4$$ $$( 1 + T )^{2}$$
$$7$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 11 T^{2} )$$

## Galois representations

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

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.90.3
$$3$$ 3.720.4

## 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

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 154.a

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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