Properties

Label 65520.a.65520.1
Conductor 65520
Discriminant -65520
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 - 28x^4 - 336x^2 - 1365$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - 28x^4z^2 - 336x^2z^4 - 1365z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 110x^4 - 1343x^2 - 5460$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1365, 0, -336, 0, -28, 0, -1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1365, 0, -336, 0, -28, 0, -1]), R([0, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-5460, 0, -1343, 0, -110, 0, -3]))
 

Invariants

Conductor: \( N \)  =  \(65520\) = \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-65520\) = \( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-6294880\) =  \( - 2^{5} \cdot 5 \cdot 39343 \)
\( I_4 \)  = \(1537408\) =  \( 2^{7} \cdot 12011 \)
\( I_6 \)  = \(-3225599450880\) =  \( - 2^{8} \cdot 3 \cdot 5 \cdot 839999857 \)
\( I_{10} \)  = \(-268369920\) =  \( - 2^{16} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( J_2 \)  = \(-786860\) =  \( - 2^{2} \cdot 5 \cdot 39343 \)
\( J_4 \)  = \(25797844802\) =  \( 2 \cdot 31 \cdot 149 \cdot 179 \cdot 15601 \)
\( J_6 \)  = \(-1127737053817920\) =  \( - 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 1553 \cdot 2770783 \)
\( J_8 \)  = \(55460595434772527999\) =  \( 463 \cdot 272462809 \cdot 439639097 \)
\( J_{10} \)  = \(-65520\) =  \( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( g_1 \)  = \(3770486200298841428197720000/819\)
\( g_2 \)  = \(157103494153138593316681400/819\)
\( g_3 \)  = \(10656851118019203761600\)

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

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

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable except over $\R$ and $\Q_{3}$.

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: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 13z^2\) \(=\) \(0,\) \(y\) \(=\) \(6xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 + 12z^2\) \(=\) \(0,\) \(2y\) \(=\) \(11xz^2\) \(0\) \(2\)

2-torsion field: 8.0.888731494560000.75

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(5\)
Regulator: \( 1 \)
Real period: \( 4.128135 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 2.064067 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(4\) \(1\) \(1 - T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T^{2} )\)
\(13\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 13 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 48.a4
  Elliptic curve 1365.e3

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 \(\Q\).