Properties

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

$y^2 + (x^3 + x)y = -x^6 - 9x^4 - 32x^2 - 39$ (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)

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

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

\( I_2 \)  = \(-181408\) =  \( - 2^{5} \cdot 5669 \)
\( I_4 \)  = \(179584\) =  \( 2^{7} \cdot 23 \cdot 61 \)
\( I_6 \)  = \(-10842583296\) =  \( - 2^{8} \cdot 3 \cdot 14117947 \)
\( I_{10} \)  = \(-7667712\) =  \( - 2^{16} \cdot 3^{2} \cdot 13 \)
\( J_2 \)  = \(-22676\) =  \( - 2^{2} \cdot 5669 \)
\( J_4 \)  = \(21423170\) =  \( 2 \cdot 5 \cdot 29 \cdot 31 \cdot 2383 \)
\( J_6 \)  = \(-26983749312\) =  \( - 2^{6} \cdot 3^{2} \cdot 13 \cdot 1069 \cdot 3371 \)
\( J_8 \)  = \(38232821637503\) =  \( 38232821637503 \)
\( J_{10} \)  = \(-1872\) =  \( - 2^{4} \cdot 3^{2} \cdot 13 \)
\( g_1 \)  = \(374724646811252438336/117\)
\( g_2 \)  = \(15612163699641478120/117\)
\( g_3 \)  = \(7411896491650496\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

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
\(2\) \(4\) \(3\) \(2\) \(1 - T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(13\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 13 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 24.a4
  Elliptic curve 39.a3

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 \(\Q\).