Properties

Label 2156.a.17248.1
Conductor $2156$
Discriminant $17248$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = 2x^5 + 5x^4 + 7x^3 + 5x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 2x^5z + 5x^4z^2 + 7x^3z^3 + 5x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 8x^5 + 21x^4 + 30x^3 + 21x^2 + 8x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 5, 7, 5, 2]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 5, 7, 5, 2], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([0, 8, 21, 30, 21, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2156\) \(=\) \( 2^{2} \cdot 7^{2} \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(17248\) \(=\) \( 2^{5} \cdot 7^{2} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(452\) \(=\)  \( 2^{2} \cdot 113 \)
\( I_4 \)  \(=\) \(24145\) \(=\)  \( 5 \cdot 11 \cdot 439 \)
\( I_6 \)  \(=\) \(4532153\) \(=\)  \( 4532153 \)
\( I_{10} \)  \(=\) \(2207744\) \(=\)  \( 2^{12} \cdot 7^{2} \cdot 11 \)
\( J_2 \)  \(=\) \(113\) \(=\)  \( 113 \)
\( J_4 \)  \(=\) \(-474\) \(=\)  \( - 2 \cdot 3 \cdot 79 \)
\( J_6 \)  \(=\) \(-28028\) \(=\)  \( - 2^{2} \cdot 7^{2} \cdot 11 \cdot 13 \)
\( J_8 \)  \(=\) \(-847960\) \(=\)  \( - 2^{3} \cdot 5 \cdot 17 \cdot 29 \cdot 43 \)
\( J_{10} \)  \(=\) \(17248\) \(=\)  \( 2^{5} \cdot 7^{2} \cdot 11 \)
\( g_1 \)  \(=\) \(18424351793/17248\)
\( g_2 \)  \(=\) \(-341966589/8624\)
\( g_3 \)  \(=\) \(-165997/8\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![0,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 + 3xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 + 3xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 + 3xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2\) \(0\) \(6\)

2-torsion field: 4.0.4312.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 7.251618 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.402867 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(5\) \(2\) \(( 1 + T )^{2}\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 11 T^{2} )\)

Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.90.3
\(3\) 3.720.4

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 154.b

Endomorphisms of the Jacobian

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 \(\Q\).