# Properties

 Label 71407.a.71407.1 Conductor 71407 Discriminant -71407 Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

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

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

$y^2 + x^3y = -2x^4 - 3x^3 + x^2 + 3x + 1$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$71407$$ = $$7 \cdot 101^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-71407$$ = $$-1 \cdot 7 \cdot 101^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$416$$ = $$2^{5} \cdot 13$$ $$I_4$$ = $$54592$$ = $$2^{6} \cdot 853$$ $$I_6$$ = $$241664$$ = $$2^{12} \cdot 59$$ $$I_{10}$$ = $$-292483072$$ = $$-1 \cdot 2^{12} \cdot 7 \cdot 101^{2}$$ $$J_2$$ = $$52$$ = $$2^{2} \cdot 13$$ $$J_4$$ = $$-456$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 19$$ $$J_6$$ = $$8120$$ = $$2^{3} \cdot 5 \cdot 7 \cdot 29$$ $$J_8$$ = $$53576$$ = $$2^{3} \cdot 37 \cdot 181$$ $$J_{10}$$ = $$-71407$$ = $$-1 \cdot 7 \cdot 101^{2}$$ $$g_1$$ = $$-380204032/71407$$ $$g_2$$ = $$64117248/71407$$ $$g_3$$ = $$-3136640/10201$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## Rational points

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);

This curve is locally solvable everywhere.

magma: [C![-7,55,3],C![-7,55,4],C![-7,288,3],C![-7,288,4],C![-1,0,1],C![-1,0,2],C![-1,1,1],C![-1,1,2],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]];

Known rational points: (-7 : 55 : 3), (-7 : 55 : 4), (-7 : 288 : 3), (-7 : 288 : 4), (-1 : 0 : 1), (-1 : 0 : 2), (-1 : 1 : 1), (-1 : 1 : 2), (0 : -1 : 1), (0 : 1 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$3$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0371760254941 Real period: 19.804710169870575377745192846 Tamagawa numbers: 1 (p = 7), 1 (p = 101) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 101.a1
Elliptic curve 707.a1

### Endomorphisms

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 $$\Q$$.