↑

Properties

Label 3468.b.353736.1
Conductor $3468$
Discriminant $353736$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{6}\Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 + 11x^4 + 27x^3 + x^2 - 34x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 11x^4z^2 + 27x^3z^3 + x^2z^4 - 34xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 45x^4 + 110x^3 + 5x^2 - 136x$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -34, 1, 27, 11, 1]), R([0, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -34, 1, 27, 11, 1], R![0, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([0, -136, 5, 110, 45, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3468\) \(=\) \( 2^{2} \cdot 3 \cdot 17^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(353736\) \(=\) \( 2^{3} \cdot 3^{2} \cdot 17^{3} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(23620\) \(=\)  \( 2^{2} \cdot 5 \cdot 1181 \)
\( I_4 \)  \(=\) \(25616905\) \(=\)  \( 5 \cdot 157 \cdot 32633 \)
\( I_6 \)  \(=\) \(160250062485\) \(=\)  \( 3 \cdot 5 \cdot 19 \cdot 12721 \cdot 44201 \)
\( I_{10} \)  \(=\) \(45278208\) \(=\)  \( 2^{10} \cdot 3^{2} \cdot 17^{3} \)
\( J_2 \)  \(=\) \(5905\) \(=\)  \( 5 \cdot 1181 \)
\( J_4 \)  \(=\) \(385505\) \(=\)  \( 5 \cdot 77101 \)
\( J_6 \)  \(=\) \(1713745\) \(=\)  \( 5 \cdot 11 \cdot 31159 \)
\( J_8 \)  \(=\) \(-34623610200\) \(=\)  \( - 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 109^{2} \cdot 1619 \)
\( J_{10} \)  \(=\) \(353736\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 17^{3} \)
\( g_1 \)  \(=\) \(7179587780780940625/353736\)
\( g_2 \)  \(=\) \(79376093464900625/353736\)
\( g_3 \)  \(=\) \(59756617248625/353736\)

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

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

Number of rational Weierstrass points: \(2\)

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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 10xz + 17z^2\) \(=\) \(0,\) \(2y\) \(=\) \(9xz^2 + 17z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 - 7xz - 34z^2\) \(=\) \(0,\) \(8y\) \(=\) \(83xz^2 + 170z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 10xz + 17z^2\) \(=\) \(0,\) \(2y\) \(=\) \(9xz^2 + 17z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 - 7xz - 34z^2\) \(=\) \(0,\) \(8y\) \(=\) \(83xz^2 + 170z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 10xz + 17z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^2z + 19xz^2 + 34z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 - 7xz - 34z^2\) \(=\) \(0,\) \(8y\) \(=\) \(x^2z + 167xz^2 + 340z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{17})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 12.52521 \)
Tamagawa product: \( 8 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.695845 \)
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?
\(2\) \(2\) \(3\) \(2\) \(-1^*\) \(( 1 - T )( 1 + T )\) yes
\(3\) \(1\) \(2\) \(2\) \(-1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) yes
\(17\) \(2\) \(3\) \(2\) \(1\) \(( 1 + T )^{2}\) 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?
\(2\) 2.180.3 yes
\(3\) 3.640.2 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 102.b
  Elliptic curve isogeny class 34.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 \(3\) 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);