Properties

Label 196.a.21952.1
Conductor 196
Discriminant -21952
Mordell-Weil group \(\Z/{6}\Z \times \Z/{6}\Z\)
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

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 + 3x^5z + 6x^4z^2 + 7x^3z^3 + 6x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 + 12x^5 + 25x^4 + 30x^3 + 25x^2 + 12x + 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 6, 7, 6, 3, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 6, 7, 6, 3, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 12, 25, 30, 25, 12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $D_6$
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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1)\)

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]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(6\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(3x^2 + 5xz + 4z^2\) \(=\) \(0,\) \(9y\) \(=\) \(8xz^2 + 13z^3\) \(0\) \(6\)

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 11.77714 \)
Tamagawa product: \( 12 \)
Torsion order:\( 36 \)
Leading coefficient: \( 0.109047 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

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

Endomorphisms of the Jacobian

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 \(\Q\).