Properties

Label 2156.b.34496.1
Conductor $2156$
Discriminant $34496$
Mordell-Weil group \(\Z \oplus \Z/{6}\Z\)
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

$y^2 + (x^2 + x)y = 2x^5 - x^4 - 5x^3 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 2x^5z - x^4z^2 - 5x^3z^3 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 8x^5 - 3x^4 - 18x^3 + x^2 + 12x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 0, -5, -1, 2]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 0, -5, -1, 2], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 12, 1, -18, -3, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2156\) \(=\) \( 2^{2} \cdot 7^{2} \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(34496\) \(=\) \( 2^{6} \cdot 7^{2} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2916\) \(=\)  \( 2^{2} \cdot 3^{6} \)
\( I_4 \)  \(=\) \(41745\) \(=\)  \( 3 \cdot 5 \cdot 11^{2} \cdot 23 \)
\( I_6 \)  \(=\) \(37024569\) \(=\)  \( 3^{2} \cdot 563 \cdot 7307 \)
\( I_{10} \)  \(=\) \(4415488\) \(=\)  \( 2^{13} \cdot 7^{2} \cdot 11 \)
\( J_2 \)  \(=\) \(729\) \(=\)  \( 3^{6} \)
\( J_4 \)  \(=\) \(20404\) \(=\)  \( 2^{2} \cdot 5101 \)
\( J_6 \)  \(=\) \(734800\) \(=\)  \( 2^{4} \cdot 5^{2} \cdot 11 \cdot 167 \)
\( J_8 \)  \(=\) \(29836496\) \(=\)  \( 2^{4} \cdot 17 \cdot 43 \cdot 2551 \)
\( J_{10} \)  \(=\) \(34496\) \(=\)  \( 2^{6} \cdot 7^{2} \cdot 11 \)
\( g_1 \)  \(=\) \(205891132094649/34496\)
\( g_2 \)  \(=\) \(1976231914389/8624\)
\( g_3 \)  \(=\) \(2218766175/196\)

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
\((1 : 0 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((-1 : 0 : 2)\)
\((1 : -2 : 1)\) \((-1 : 2 : 2)\)
All points
\((1 : 0 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((-1 : 0 : 2)\)
\((1 : -2 : 1)\) \((-1 : 2 : 2)\)
All points
\((1 : 0 : 0)\) \((-1 : 0 : 1)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\)
\((-1 : -2 : 2)\) \((-1 : 2 : 2)\)

magma: [C![-1,0,1],C![-1,0,2],C![-1,2,2],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-1,0,1],C![-1,-2,2],C![-1,2,2],C![0,-2,1],C![0,2,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // 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: \(\Z \oplus \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.123038\) \(\infty\)
\((-1 : 2 : 2) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-3xz^2 - z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.123038\) \(\infty\)
\((-1 : 2 : 2) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-3xz^2 - z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((0 : -2 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - 2z^3\) \(0.123038\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(x^2z - 5xz^2 - 2z^3\) \(0\) \(6\)

2-torsion field: 4.0.2156.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.123038 \)
Real period: \( 21.05583 \)
Tamagawa product: \( 4 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.287853 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(6\) \(4\) \(( 1 + T )^{2}\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 11 T^{2} )\)

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.a

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 \(\Q\).