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Properties

Label 20412.b.734832.1
Conductor $20412$
Discriminant $734832$
Mordell-Weil group \(\Z/{3}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^6 + 3x^5 + 2x^4 + 7x^3 + 11x^2 + 14$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 + 3x^5z + 2x^4z^2 + 7x^3z^3 + 11x^2z^4 + 14z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 + 12x^5 + 9x^4 + 30x^3 + 45x^2 + 56$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([14, 0, 11, 7, 2, 3, 1]), R([0, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![14, 0, 11, 7, 2, 3, 1], R![0, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([56, 0, 45, 30, 9, 12, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(20412\) \(=\) \( 2^{2} \cdot 3^{6} \cdot 7 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(734832\) \(=\) \( 2^{4} \cdot 3^{8} \cdot 7 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(9140\) \(=\)  \( 2^{2} \cdot 5 \cdot 457 \)
\( I_4 \)  \(=\) \(5234697\) \(=\)  \( 3^{2} \cdot 13 \cdot 44741 \)
\( I_6 \)  \(=\) \(11956156581\) \(=\)  \( 3 \cdot 19 \cdot 209757133 \)
\( I_{10} \)  \(=\) \(-387072\) \(=\)  \( - 2^{11} \cdot 3^{3} \cdot 7 \)
\( J_2 \)  \(=\) \(6855\) \(=\)  \( 3 \cdot 5 \cdot 457 \)
\( J_4 \)  \(=\) \(-5052\) \(=\)  \( - 2^{2} \cdot 3 \cdot 421 \)
\( J_6 \)  \(=\) \(-1696\) \(=\)  \( - 2^{5} \cdot 53 \)
\( J_8 \)  \(=\) \(-9287196\) \(=\)  \( - 2^{2} \cdot 3 \cdot 773933 \)
\( J_{10} \)  \(=\) \(-734832\) \(=\)  \( - 2^{4} \cdot 3^{8} \cdot 7 \)
\( g_1 \)  \(=\) \(-62291820293928125/3024\)
\( g_2 \)  \(=\) \(5022740131625/2268\)
\( g_3 \)  \(=\) \(553449850/5103\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : -1 : 0),\, (1 : 1 : 0)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0)\)
All points: \((1 : -2 : 0),\, (1 : 2 : 0)\)

Copy content magma:[C![1,-1,0],C![1,1,0]]; // minimal model
 
Copy content magma:[C![1,-2,0],C![1,2,0]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{3}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.2.28449792.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 4.793478 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.532608 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(4\) \(1\) \(1^*\) \(( 1 + T )^{2}\) yes
\(3\) \(6\) \(8\) \(1\) \(1^*\) \(1\) no
\(7\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 7 T^{2} )\) yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.20.2 no
\(3\) 3.720.4 yes

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 54.a
  Elliptic curve isogeny class 378.a

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(7\) 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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);