Properties

Label 3645.a.295245.1
Conductor $3645$
Discriminant $295245$
Mordell-Weil group \(\Z \oplus \Z/{3}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = x^5 - 7x^3 + x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^5z - 7x^3z^3 + xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 6x^5 + 3x^4 - 24x^3 + 3x^2 + 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -7, 0, 1]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -7, 0, 1], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, 3, -24, 3, 6, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3645\) \(=\) \( 3^{6} \cdot 5 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(295245\) \(=\) \( 3^{10} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(376\) \(=\)  \( 2^{3} \cdot 47 \)
\( I_4 \)  \(=\) \(7380\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 41 \)
\( I_6 \)  \(=\) \(710544\) \(=\)  \( 2^{4} \cdot 3 \cdot 113 \cdot 131 \)
\( I_{10} \)  \(=\) \(4860\) \(=\)  \( 2^{2} \cdot 3^{5} \cdot 5 \)
\( J_2 \)  \(=\) \(564\) \(=\)  \( 2^{2} \cdot 3 \cdot 47 \)
\( J_4 \)  \(=\) \(2184\) \(=\)  \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
\( J_6 \)  \(=\) \(17960\) \(=\)  \( 2^{3} \cdot 5 \cdot 449 \)
\( J_8 \)  \(=\) \(1339896\) \(=\)  \( 2^{3} \cdot 3 \cdot 55829 \)
\( J_{10} \)  \(=\) \(295245\) \(=\)  \( 3^{10} \cdot 5 \)
\( g_1 \)  \(=\) \(234849287168/1215\)
\( g_2 \)  \(=\) \(4837321216/3645\)
\( g_3 \)  \(=\) \(126955648/6561\)

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.059091\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.059091\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 + z^3\) \(0.059091\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 - z^3\) \(0\) \(3\)

2-torsion field: 6.2.233280.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.059091 \)
Real period: \( 20.04123 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.394754 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(6\) \(10\) \(3\) \(1\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 5 T^{2} )\)

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?
\(2\) 2.15.2 no
\(3\) 3.720.4 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

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 135.a
  Elliptic curve isogeny class 27.a

magma: HeuristicDecompositionFactors(C);
 

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 order of index \(12\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)

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

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

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