Properties

Label 23412.a.140472.1
Conductor $23412$
Discriminant $-140472$
Mordell-Weil group \(\Z \oplus \Z\)
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

$y^2 + (x^3 + x + 1)y = 2x^4 + x^3 - 2x^2 - x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = 2x^4z^2 + x^3z^3 - 2x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 10x^4 + 6x^3 - 7x^2 - 2x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 1, 2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 1, 2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, -7, 6, 10, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(23412\) \(=\) \( 2^{2} \cdot 3 \cdot 1951 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-140472\) \(=\) \( - 2^{3} \cdot 3^{2} \cdot 1951 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(548\) \(=\)  \( 2^{2} \cdot 137 \)
\( I_4 \)  \(=\) \(27745\) \(=\)  \( 5 \cdot 31 \cdot 179 \)
\( I_6 \)  \(=\) \(3116593\) \(=\)  \( 17 \cdot 183329 \)
\( I_{10} \)  \(=\) \(-17980416\) \(=\)  \( - 2^{10} \cdot 3^{2} \cdot 1951 \)
\( J_2 \)  \(=\) \(137\) \(=\)  \( 137 \)
\( J_4 \)  \(=\) \(-374\) \(=\)  \( - 2 \cdot 11 \cdot 17 \)
\( J_6 \)  \(=\) \(6660\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 \)
\( J_8 \)  \(=\) \(193136\) \(=\)  \( 2^{4} \cdot 12071 \)
\( J_{10} \)  \(=\) \(-140472\) \(=\)  \( - 2^{3} \cdot 3^{2} \cdot 1951 \)
\( g_1 \)  \(=\) \(-48261724457/140472\)
\( g_2 \)  \(=\) \(480843011/70236\)
\( g_3 \)  \(=\) \(-3472265/3902\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 1 : 1)\) \((-1 : 0 : 2)\) \((1 : -3 : 1)\) \((-1 : -3 : 2)\) \((-6 : -10 : 1)\) \((2 : -18 : 3)\)
\((2 : -35 : 3)\) \((-6 : 231 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 1 : 1)\) \((-1 : 0 : 2)\) \((1 : -3 : 1)\) \((-1 : -3 : 2)\) \((-6 : -10 : 1)\) \((2 : -18 : 3)\)
\((2 : -35 : 3)\) \((-6 : 231 : 1)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -3 : 1)\) \((1 : 3 : 1)\) \((-1 : -3 : 2)\) \((-1 : 3 : 2)\) \((2 : -17 : 3)\) \((2 : 17 : 3)\)
\((-6 : -241 : 1)\) \((-6 : 241 : 1)\)

magma: [C![-6,-10,1],C![-6,231,1],C![-1,-3,2],C![-1,0,1],C![-1,0,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-35,3],C![2,-18,3]]; // minimal model
 
magma: [C![-6,-241,1],C![-6,241,1],C![-1,-3,2],C![-1,-1,1],C![-1,3,2],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![2,-17,3],C![2,17,3]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.146123\) \(\infty\)
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.044858\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.146123\) \(\infty\)
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.044858\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.146123\) \(\infty\)
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 - z^3\) \(0.044858\) \(\infty\)

2-torsion field: 6.4.998912.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006487 \)
Real period: \( 19.45348 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.757176 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(1951\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 69 T + 1951 T^{2} )\)

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?
\(2\) 2.10.1 no

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

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

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);