Properties

Label 196.a.21952.1
Conductor 196
Discriminant -21952
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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This is a model for the modular curve $X_0(28)$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 6, 7, 6, 3, 1], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 6, 7, 6, 3, 1]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 196 \)  =  \( 2^{2} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-21952\)  =  \( -1 \cdot 2^{6} \cdot 7^{3} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(-2680\)  =  \( -1 \cdot 2^{3} \cdot 5 \cdot 67 \)
\( I_4 \)  =  \(5380\)  =  \( 2^{2} \cdot 5 \cdot 269 \)
\( I_6 \)  =  \(-1198840\)  =  \( -1 \cdot 2^{3} \cdot 5 \cdot 17 \cdot 41 \cdot 43 \)
\( I_{10} \)  =  \(-89915392\)  =  \( -1 \cdot 2^{18} \cdot 7^{3} \)
\( J_2 \)  =  \(-335\)  =  \( -1 \cdot 5 \cdot 67 \)
\( J_4 \)  =  \(4620\)  =  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
\( J_6 \)  =  \(-90160\)  =  \( -1 \cdot 2^{4} \cdot 5 \cdot 7^{2} \cdot 23 \)
\( J_8 \)  =  \(2214800\)  =  \( 2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 113 \)
\( J_{10} \)  =  \(-21952\)  =  \( -1 \cdot 2^{6} \cdot 7^{3} \)
\( g_1 \)  =  \(4219140959375/21952\)
\( g_2 \)  =  \(6203236875/784\)
\( g_3 \)  =  \(12905875/28\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(D_6 \) (GAP id : [12,4])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_6 \) (GAP id : [12,4])

Rational points

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);

This curve is locally solvable everywhere.

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

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

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 4 (p = 2), 3 (p = 7)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: \(\Z/{6}\Z \times \Z/{6}\Z\)

2-torsion field: \(\Q(\sqrt{-7}) \)

Sato-Tate group

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

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 14.a6

Endomorphisms

Not of \(\GL_2\)-type over \(\Q\)

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).