Properties

Label 65520.b.131040.1
Conductor 65520
Discriminant -131040
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
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

$y^2 + (x^3 + x)y = -10x^6 - 82x^4 - 227x^2 - 210$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -10x^6 - 82x^4z^2 - 227x^2z^4 - 210z^6$ (dehomogenize, simplify)
$y^2 = -39x^6 - 326x^4 - 907x^2 - 840$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-210, 0, -227, 0, -82, 0, -10], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-210, 0, -227, 0, -82, 0, -10]), R([0, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-840, 0, -907, 0, -326, 0, -39]))
 

Invariants

Conductor: \( N \)  =  \(65520\) = \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-131040\) = \( - 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-12593312\) =  \( - 2^{5} \cdot 393541 \)
\( I_4 \)  = \(11343232\) =  \( 2^{7} \cdot 23 \cdot 3853 \)
\( I_6 \)  = \(-47604155826432\) =  \( - 2^{8} \cdot 3 \cdot 97 \cdot 9431 \cdot 67757 \)
\( I_{10} \)  = \(-536739840\) =  \( - 2^{17} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( J_2 \)  = \(-1574164\) =  \( - 2^{2} \cdot 393541 \)
\( J_4 \)  = \(103249560962\) =  \( 2 \cdot 17 \cdot 83 \cdot 5927 \cdot 6173 \)
\( J_6 \)  = \(-9029520569946240\) =  \( - 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31 \cdot 555698369 \)
\( J_8 \)  = \(888368594905774644479\) =  \( 888368594905774644479 \)
\( J_{10} \)  = \(-131040\) =  \( - 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( g_1 \)  = \(302064649214662608101539958432/4095\)
\( g_2 \)  = \(12586012647194024913614166004/4095\)
\( g_3 \)  = \(170750018582492394877376\)

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 $\R$.

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(8x^2 + 21z^2\) \(=\) \(0,\) \(8y\) \(=\) \(7xz^2\) \(0.532646\) \(\infty\)
\(D_0 - D_\infty\) \(3x^2 + 8z^2\) \(=\) \(0,\) \(6y\) \(=\) \(5xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(2\)

2-torsion field: splitting field of \(x^{8} - 4 x^{7} + 466 x^{6} - 1384 x^{5} + 79827 x^{4} - 157352 x^{3} + 5948566 x^{2} - 5870120 x + 163148164\) with Galois group $C_2^3$

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(6\)
Regulator: \( 0.532646 \)
Real period: \( 3.186149 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.697091 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

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

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 312.f2
  Elliptic curve 210.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 \(\Q\).