# Properties

 Label 100352.c.100352.1 Conductor 100352 Discriminant 100352 Mordell-Weil group $$\Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 = x^5 - 9x^3 + 7x^2 + 14x - 14$ (homogenize, simplify) $y^2 = x^5z - 9x^3z^3 + 7x^2z^4 + 14xz^5 - 14z^6$ (dehomogenize, simplify) $y^2 = x^5 - 9x^3 + 7x^2 + 14x - 14$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-14, 14, 7, -9, 0, 1], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, 14, 7, -9, 0, 1]), R([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-14, 14, 7, -9, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$16736$$ = $$2^{5} \cdot 523$$ $$I_4$$ = $$265216$$ = $$2^{10} \cdot 7 \cdot 37$$ $$I_6$$ = $$1458946048$$ = $$2^{14} \cdot 7 \cdot 12721$$ $$I_{10}$$ = $$411041792$$ = $$2^{23} \cdot 7^{2}$$ $$J_2$$ = $$2092$$ = $$2^{2} \cdot 523$$ $$J_4$$ = $$179590$$ = $$2 \cdot 5 \cdot 17959$$ $$J_6$$ = $$20265956$$ = $$2^{2} \cdot 71 \cdot 71359$$ $$J_8$$ = $$2535952963$$ = $$2535952963$$ $$J_{10}$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ $$g_1$$ = $$39129873538843/98$$ $$g_2$$ = $$12845683618265/784$$ $$g_3$$ = $$1385831669681/1568$$

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

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

## 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

All points: $$(1 : 0 : 0),\, (9 : -182 : 4),\, (9 : 182 : 4)$$

magma: [C![1,0,0],C![9,-182,4],C![9,182,4]];

Number of rational Weierstrass points: $$1$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + 2z^3$$ $$1.026920$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$1.026920$$ Real period: $$10.31998$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$2.649451$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

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

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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