Properties

Label 1270.a.325120.1
Conductor $1270$
Discriminant $325120$
Mordell-Weil group \(\Z/{2}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 + 17x^4 + 76x^3 + 14x^2 - 32x + 3$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 17x^4z^2 + 76x^3z^3 + 14x^2z^4 - 32xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 69x^4 + 306x^3 + 57x^2 - 128x + 12$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([3, -32, 14, 76, 17, 1]), R([0, 1, 1])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![3, -32, 14, 76, 17, 1], R![0, 1, 1]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([12, -128, 57, 306, 69, 4])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(1270\) \(=\) \( 2 \cdot 5 \cdot 127 \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(325120\) \(=\) \( 2^{9} \cdot 5 \cdot 127 \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(239204\) \(=\)  \( 2^{2} \cdot 7 \cdot 8543 \)
\( I_4 \)  \(=\) \(126763297\) \(=\)  \( 126763297 \)
\( I_6 \)  \(=\) \(10436094933809\) \(=\)  \( 19 \cdot 87961 \cdot 6244451 \)
\( I_{10} \)  \(=\) \(41615360\) \(=\)  \( 2^{16} \cdot 5 \cdot 127 \)
\( J_2 \)  \(=\) \(59801\) \(=\)  \( 7 \cdot 8543 \)
\( J_4 \)  \(=\) \(143724846\) \(=\)  \( 2 \cdot 3 \cdot 23954141 \)
\( J_6 \)  \(=\) \(437833820176\) \(=\)  \( 2^{4} \cdot 1009 \cdot 27120529 \)
\( J_8 \)  \(=\) \(1381517230655315\) \(=\)  \( 5 \cdot 13 \cdot 19 \cdot 1373^{2} \cdot 593401 \)
\( J_{10} \)  \(=\) \(325120\) \(=\)  \( 2^{9} \cdot 5 \cdot 127 \)
\( g_1 \)  \(=\) \(764790054928595680699001/325120\)
\( g_2 \)  \(=\) \(15368348330455841308623/162560\)
\( g_3 \)  \(=\) \(97860226229056869361/20320\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:Cmin.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(Cmin); IgusaInvariants(Cmin); G2Invariants(Cmin);
 

Automorphism group

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

Rational points

All points: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)

Copy content magma:[Cmin![1,0,0]]; // minimal model
 
Copy content magma:[Csim![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(Cmin)));
 

This curve is locally solvable everywhere.

Copy content magma:f,h:=HyperellipticPolynomials(Cmin); 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: \(\Z/{2}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(Cmin));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 8xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(7xz^2 - 4z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 8xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(7xz^2 - 4z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 8xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(2x^2z + 16xz^2 - 8z^3\) \(0\) \(2\)

2-torsion field: 6.6.129032000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 1.894603 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.473650 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(1\) \(9\) \(1\) \(1^*\) \(( 1 + T )( 1 - T + 2 T^{2} )\) yes
\(5\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 3 T + 5 T^{2} )\) yes
\(127\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 - 20 T + 127 T^{2} )\) yes

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes
\(3\) 3.80.2 no

Sato-Tate group

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

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Copy content magma:HeuristicDecompositionFactors(Cmin);
 

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 \(\Q\).

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

Copy content magma:HeuristicIsGL2(Cmin); HeuristicEndomorphismDescription(Cmin); HeuristicEndomorphismFieldOfDefinition(Cmin);
 

Copy content magma:HeuristicIsGL2(Cmin : Geometric := true); HeuristicEndomorphismDescription(Cmin : Geometric := true); HeuristicEndomorphismLatticeDescription(Cmin);