Properties

Label 1320.a.2640.1
Conductor $1320$
Discriminant $2640$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{4}\Z\)
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

$y^2 + (x^3 + x)y = -x^6 + 9x^4 - 40x^2 + 55$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 + 9x^4z^2 - 40x^2z^4 + 55z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 38x^4 - 159x^2 + 220$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([55, 0, -40, 0, 9, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![55, 0, -40, 0, 9, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([220, 0, -159, 0, 38, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1320\) \(=\) \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(2640\) \(=\) \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(63768\) \(=\)  \( 2^{3} \cdot 3 \cdot 2657 \)
\( I_4 \)  \(=\) \(10392\) \(=\)  \( 2^{3} \cdot 3 \cdot 433 \)
\( I_6 \)  \(=\) \(220729308\) \(=\)  \( 2^{2} \cdot 3 \cdot 19 \cdot 968111 \)
\( I_{10} \)  \(=\) \(10560\) \(=\)  \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \)
\( J_2 \)  \(=\) \(31884\) \(=\)  \( 2^{2} \cdot 3 \cdot 2657 \)
\( J_4 \)  \(=\) \(42356162\) \(=\)  \( 2 \cdot 21178081 \)
\( J_6 \)  \(=\) \(75020763840\) \(=\)  \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 31 \cdot 101 \cdot 2269 \)
\( J_8 \)  \(=\) \(149479393726079\) \(=\)  \( 16349 \cdot 9143029771 \)
\( J_{10} \)  \(=\) \(2640\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
\( g_1 \)  \(=\) \(686471900571962215488/55\)
\( g_2 \)  \(=\) \(28601826290311163976/55\)
\( g_3 \)  \(=\) \(28888377841215936\)

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: \((-2 : 5 : 1),\, (2 : -5 : 1)\)
All points: \((-2 : 5 : 1),\, (2 : -5 : 1)\)
All points: \((-2 : 0 : 1),\, (2 : 0 : 1)\)

magma: [C![-2,5,1],C![2,-5,1]]; // minimal model
 
magma: [C![-2,0,1],C![2,0,1]]; // 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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 9xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 + 2z^3\) \(0\) \(4\)

2-torsion field: \(\Q(\sqrt{5}, \sqrt{33})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 17.74674 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.554585 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(4\) \(2\) \(1 - T + 2 T^{2}\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 11 T^{2} )\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

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

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.180.3 yes
\(3\) 3.90.1 no

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 24.a
  Elliptic curve isogeny class 55.a

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

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);