Properties

Label 1300.a.130000.1
Conductor 1300
Discriminant -130000
Mordell-Weil group \(\Z \times \Z/{6}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = 2x^4 + 9x^2 + 13$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 2x^4z^2 + 9x^2z^4 + 13z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 10x^4 + 37x^2 + 52$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([13, 0, 9, 0, 2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![13, 0, 9, 0, 2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([52, 0, 37, 0, 10, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1300\) = \( 2^{2} \cdot 5^{2} \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-130000\) = \( - 2^{4} \cdot 5^{4} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-18400\) =  \( - 2^{5} \cdot 5^{2} \cdot 23 \)
\( I_4 \)  = \(158464\) =  \( 2^{8} \cdot 619 \)
\( I_6 \)  = \(-968970496\) =  \( - 2^{8} \cdot 13 \cdot 23 \cdot 12659 \)
\( I_{10} \)  = \(-532480000\) =  \( - 2^{16} \cdot 5^{4} \cdot 13 \)
\( J_2 \)  = \(-2300\) =  \( - 2^{2} \cdot 5^{2} \cdot 23 \)
\( J_4 \)  = \(218766\) =  \( 2 \cdot 3 \cdot 19^{2} \cdot 101 \)
\( J_6 \)  = \(-27536704\) =  \( - 2^{6} \cdot 13 \cdot 23 \cdot 1439 \)
\( J_8 \)  = \(3868964111\) =  \( 67 \cdot 1033 \cdot 55901 \)
\( J_{10} \)  = \(-130000\) =  \( - 2^{4} \cdot 5^{4} \cdot 13 \)
\( g_1 \)  = \(6436343000000/13\)
\( g_2 \)  = \(266172592200/13\)
\( g_3 \)  = \(1120532032\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -4 : 1),\, (1 : 4 : 1),\, (-1 : 6 : 1),\, (1 : -6 : 1)\)

magma: [C![-1,-4,1],C![-1,6,1],C![1,-6,1],C![1,-1,0],C![1,0,0],C![1,4,1]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz + 9z^2\) \(=\) \(0,\) \(4y\) \(=\) \(5xz^2 - z^3\) \(0.093878\) \(\infty\)
\((-1 : -4 : 1) + (1 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 5z^3\) \(0\) \(6\)

2-torsion field: 4.0.832.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.093878 \)
Real period: \( 7.601598 \)
Tamagawa product: \( 12 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.237875 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 20.a4
  Elliptic curve 65.a2

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 \(\Q\).