Properties

Label 504.a.27216.1
Conductor $504$
Discriminant $-27216$
Mordell-Weil group \(\Z/{4}\Z \times \Z/{4}\Z\)
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

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Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = 3x^4 + 15x^2 + 21$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 3x^4z^2 + 15x^2z^4 + 21z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 14x^4 + 61x^2 + 84$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([21, 0, 15, 0, 3]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![21, 0, 15, 0, 3], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([84, 0, 61, 0, 14, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(504\) \(=\) \( 2^{3} \cdot 3^{2} \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-27216\) \(=\) \( - 2^{4} \cdot 3^{5} \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8456\) \(=\)  \( 2^{3} \cdot 7 \cdot 151 \)
\( I_4 \)  \(=\) \(9496\) \(=\)  \( 2^{3} \cdot 1187 \)
\( I_6 \)  \(=\) \(26675348\) \(=\)  \( 2^{2} \cdot 7 \cdot 952691 \)
\( I_{10} \)  \(=\) \(108864\) \(=\)  \( 2^{6} \cdot 3^{5} \cdot 7 \)
\( J_2 \)  \(=\) \(4228\) \(=\)  \( 2^{2} \cdot 7 \cdot 151 \)
\( J_4 \)  \(=\) \(743250\) \(=\)  \( 2 \cdot 3 \cdot 5^{3} \cdot 991 \)
\( J_6 \)  \(=\) \(173847744\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 43117 \)
\( J_8 \)  \(=\) \(45651924783\) \(=\)  \( 3^{3} \cdot 59 \cdot 191 \cdot 150041 \)
\( J_{10} \)  \(=\) \(27216\) \(=\)  \( 2^{4} \cdot 3^{5} \cdot 7 \)
\( g_1 \)  \(=\) \(12063042849801664/243\)
\( g_2 \)  \(=\) \(167186257609000/81\)
\( g_3 \)  \(=\) \(3083035208512/27\)

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: \((1 : 0 : 0),\, (1 : -1 : 0)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0)\)

magma: [C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![1,-1,0],C![1,1,0]]; // simplified model
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 8.0.49787136.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 7.782699 \)
Tamagawa product: \( 8 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.243209 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 21.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 \(\Q\).