Properties

Label 476.a.952.1
Conductor 476
Discriminant -952
Mordell-Weil group \(\Z/{3}\Z \times \Z/{6}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -5x^4 + 7x^3 + 25x^2 - 75x + 54$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -5x^4z^2 + 7x^3z^3 + 25x^2z^4 - 75xz^5 + 54z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 20x^4 + 30x^3 + 100x^2 - 300x + 217$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([54, -75, 25, 7, -5]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![54, -75, 25, 7, -5], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([217, -300, 100, 30, -20, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(476\) \(=\) \( 2^{2} \cdot 7 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-952\) \(=\) \( - 2^{3} \cdot 7 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-14680\) \(=\)  \( - 2^{3} \cdot 5 \cdot 367 \)
\( I_4 \)  \(=\) \(4169380\) \(=\)  \( 2^{2} \cdot 5 \cdot 208469 \)
\( I_6 \)  \(=\) \(-23242186840\) \(=\)  \( - 2^{3} \cdot 5 \cdot 13789 \cdot 42139 \)
\( I_{10} \)  \(=\) \(-3899392\) \(=\)  \( - 2^{15} \cdot 7 \cdot 17 \)
\( J_2 \)  \(=\) \(-1835\) \(=\)  \( - 5 \cdot 367 \)
\( J_4 \)  \(=\) \(96870\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 3229 \)
\( J_6 \)  \(=\) \(3910340\) \(=\)  \( 2^{2} \cdot 5 \cdot 7 \cdot 17 \cdot 31 \cdot 53 \)
\( J_8 \)  \(=\) \(-4139817700\) \(=\)  \( - 2^{2} \cdot 5^{2} \cdot 41398177 \)
\( J_{10} \)  \(=\) \(-952\) \(=\)  \( - 2^{3} \cdot 7 \cdot 17 \)
\( g_1 \)  \(=\) \(20805604708146875/952\)
\( g_2 \)  \(=\) \(299272981175625/476\)
\( g_3 \)  \(=\) \(-27661753375/2\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)\)

magma: [C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-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: \(\Z/{3}\Z \times \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 3xz - 12z^2\) \(=\) \(0,\) \(4y\) \(=\) \(-21xz^2 + 24z^3\) \(0\) \(3\)
\((2 : -4 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 4z^3\) \(0\) \(6\)

2-torsion field: 4.2.7616.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 26.72233 \)
Tamagawa product: \( 3 \)
Torsion order:\( 18 \)
Leading coefficient: \( 0.247429 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 17 T^{2} )\)

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 14.a4
  Elliptic curve 34.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 \(\Q\).