# Properties

 Label 10121.a.293509.1 Conductor $10121$ Discriminant $293509$ Mordell-Weil group $$\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + y = x^5 + 2x^4 + 4x + 3$ (homogenize, simplify) $y^2 + z^3y = x^5z + 2x^4z^2 + 4xz^5 + 3z^6$ (dehomogenize, simplify) $y^2 = 4x^5 + 8x^4 + 16x + 13$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 4, 0, 0, 2, 1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 4, 0, 0, 2, 1], R![1]);

sage: X = HyperellipticCurve(R([13, 16, 0, 0, 8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10121$$ $$=$$ $$29 \cdot 349$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$293509$$ $$=$$ $$29^{2} \cdot 349$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$640$$ $$=$$ $$2^{7} \cdot 5$$ $$I_4$$ $$=$$ $$-512$$ $$=$$ $$- 2^{9}$$ $$I_6$$ $$=$$ $$-234048$$ $$=$$ $$- 2^{6} \cdot 3 \cdot 23 \cdot 53$$ $$I_{10}$$ $$=$$ $$1174036$$ $$=$$ $$2^{2} \cdot 29^{2} \cdot 349$$ $$J_2$$ $$=$$ $$320$$ $$=$$ $$2^{6} \cdot 5$$ $$J_4$$ $$=$$ $$4352$$ $$=$$ $$2^{8} \cdot 17$$ $$J_6$$ $$=$$ $$94272$$ $$=$$ $$2^{6} \cdot 3 \cdot 491$$ $$J_8$$ $$=$$ $$2806784$$ $$=$$ $$2^{10} \cdot 2741$$ $$J_{10}$$ $$=$$ $$293509$$ $$=$$ $$29^{2} \cdot 349$$ $$g_1$$ $$=$$ $$3355443200000/293509$$ $$g_2$$ $$=$$ $$142606336000/293509$$ $$g_3$$ $$=$$ $$9653452800/293509$$

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

 All points: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (3 : 20 : 1),\, (3 : -21 : 1)$$ All points: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (3 : 20 : 1),\, (3 : -21 : 1)$$ All points: $$(1 : 0 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (3 : -41 : 1),\, (3 : 41 : 1)$$

magma: [C![-1,-1,1],C![-1,0,1],C![1,0,0],C![3,-21,1],C![3,20,1]]; // minimal model

magma: [C![-1,-1,1],C![-1,1,1],C![1,0,0],C![3,-41,1],C![3,41,1]]; // simplified model

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - 3z^3$$ $$0.042927$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - 3z^3$$ $$0.042927$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - 5z^3$$ $$0.042927$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.042927$$ Real period: $$10.77718$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.925276$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$29$$ $$1$$ $$2$$ $$2$$ $$( 1 - T )( 1 - 5 T + 29 T^{2} )$$
$$349$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 26 T + 349 T^{2} )$$

## 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 $$\Q$$.