Properties

Label 936.a.1872.1
Conductor $936$
Discriminant $-1872$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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

Related objects

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 - 9x^4 - 32x^2 - 39$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - 9x^4z^2 - 32x^2z^4 - 39z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 34x^4 - 127x^2 - 156$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-39, 0, -32, 0, -9, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-39, 0, -32, 0, -9, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-156, 0, -127, 0, -34, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(936\) \(=\) \( 2^{3} \cdot 3^{2} \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-1872\) \(=\) \( - 2^{4} \cdot 3^{2} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(45352\) \(=\)  \( 2^{3} \cdot 5669 \)
\( I_4 \)  \(=\) \(11224\) \(=\)  \( 2^{3} \cdot 23 \cdot 61 \)
\( I_6 \)  \(=\) \(169415364\) \(=\)  \( 2^{2} \cdot 3 \cdot 14117947 \)
\( I_{10} \)  \(=\) \(7488\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 13 \)
\( J_2 \)  \(=\) \(22676\) \(=\)  \( 2^{2} \cdot 5669 \)
\( J_4 \)  \(=\) \(21423170\) \(=\)  \( 2 \cdot 5 \cdot 29 \cdot 31 \cdot 2383 \)
\( J_6 \)  \(=\) \(26983749312\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 13 \cdot 1069 \cdot 3371 \)
\( J_8 \)  \(=\) \(38232821637503\) \(=\)  \( 38232821637503 \)
\( J_{10} \)  \(=\) \(1872\) \(=\)  \( 2^{4} \cdot 3^{2} \cdot 13 \)
\( g_1 \)  \(=\) \(374724646811252438336/117\)
\( g_2 \)  \(=\) \(15612163699641478120/117\)
\( g_3 \)  \(=\) \(7411896491650496\)

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 8.0.592240896.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 7.131061 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.445691 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

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

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 39.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 \(\Q\).