Properties

Label 2730.b.38220.1
Conductor $2730$
Discriminant $38220$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{4}\Z\)
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

$y^2 + (x^2 + x)y = x^5 + 28x^4 + 201x^3 + 28x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 28x^4z^2 + 201x^3z^3 + 28x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 113x^4 + 806x^3 + 113x^2 + 4x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 28, 201, 28, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 28, 201, 28, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, 113, 806, 113, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2730\) \(=\) \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(38220\) \(=\) \( 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1847076\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 1999 \)
\( I_4 \)  \(=\) \(9790593\) \(=\)  \( 3 \cdot 971 \cdot 3361 \)
\( I_6 \)  \(=\) \(6017800236609\) \(=\)  \( 3 \cdot 7 \cdot 97 \cdot 7349 \cdot 401993 \)
\( I_{10} \)  \(=\) \(4892160\) \(=\)  \( 2^{9} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \)
\( J_2 \)  \(=\) \(461769\) \(=\)  \( 3 \cdot 7 \cdot 11 \cdot 1999 \)
\( J_4 \)  \(=\) \(8884200782\) \(=\)  \( 2 \cdot 17 \cdot 31 \cdot 311 \cdot 27103 \)
\( J_6 \)  \(=\) \(227893017162720\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 191 \cdot 193 \cdot 20219 \)
\( J_8 \)  \(=\) \(6576226776830660039\) \(=\)  \( 14341843 \cdot 458534288573 \)
\( J_{10} \)  \(=\) \(38220\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \)
\( g_1 \)  \(=\) \(142825757240820183004850067/260\)
\( g_2 \)  \(=\) \(2975399985891326799051477/130\)
\( g_3 \)  \(=\) \(1271422473017363079336\)

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

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

Number of rational Weierstrass points: \(2\)

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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 14xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(13xz^2 + z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 14xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-113xz^2 - 8z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 14xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(13xz^2 + z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 14xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-113xz^2 - 8z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 14xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^2z + 27xz^2 + 2z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 14xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 225xz^2 - 16z^3\) \(0\) \(4\)

2-torsion field: \(\Q(\sqrt{3}, \sqrt{65})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 2.823465 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.705866 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 7 T^{2} )\)
\(13\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 13 T^{2} )\)

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?
\(2\) 2.180.3 yes
\(3\) 3.90.1 no

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 130.b
  Elliptic curve isogeny class 21.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 \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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

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);