Properties

Label 26244.c.157464.1
Conductor $26244$
Discriminant $157464$
Mordell-Weil group \(\Z/{3}\Z \times \Z/{3}\Z\)
Sato-Tate group $J(E_1)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 2x^3$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 2x^3z^3$ (dehomogenize, simplify)
$y^2 = x^6 + 10x^3 + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 0, 0, 10, 0, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(26244\) \(=\) \( 2^{2} \cdot 3^{8} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(157464\) \(=\) \( 2^{3} \cdot 3^{9} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(60\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(945\) \(=\)  \( 3^{3} \cdot 5 \cdot 7 \)
\( I_6 \)  \(=\) \(2295\) \(=\)  \( 3^{3} \cdot 5 \cdot 17 \)
\( I_{10} \)  \(=\) \(82944\) \(=\)  \( 2^{10} \cdot 3^{4} \)
\( J_2 \)  \(=\) \(45\) \(=\)  \( 3^{2} \cdot 5 \)
\( J_4 \)  \(=\) \(-270\) \(=\)  \( - 2 \cdot 3^{3} \cdot 5 \)
\( J_6 \)  \(=\) \(3780\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \)
\( J_8 \)  \(=\) \(24300\) \(=\)  \( 2^{2} \cdot 3^{5} \cdot 5^{2} \)
\( J_{10} \)  \(=\) \(157464\) \(=\)  \( 2^{3} \cdot 3^{9} \)
\( g_1 \)  \(=\) \(9375/8\)
\( g_2 \)  \(=\) \(-625/4\)
\( g_3 \)  \(=\) \(875/18\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{3}\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0\) \(3\)

2-torsion field: 6.0.1259712.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 14.14876 \)
Tamagawa product: \( 9 \)
Torsion order:\( 9 \)
Leading coefficient: \( 1.572084 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)
\(3\) \(8\) \(9\) \(3\) \(1\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_1)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 162.b4
  Elliptic curve 162.c3

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)