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Properties

Label 1920.a.368640.1
Conductor $1920$
Discriminant $-368640$
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

$y^2 + (x^3 + x^2 + x + 1)y = 5x^6 + 6x^5 + 17x^4 + 12x^3 + 17x^2 + 6x + 5$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = 5x^6 + 6x^5z + 17x^4z^2 + 12x^3z^3 + 17x^2z^4 + 6xz^5 + 5z^6$ (dehomogenize, simplify)
$y^2 = 21x^6 + 26x^5 + 71x^4 + 52x^3 + 71x^2 + 26x + 21$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 6, 17, 12, 17, 6, 5]), R([1, 1, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 6, 17, 12, 17, 6, 5], R![1, 1, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([21, 26, 71, 52, 71, 26, 21]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1920\) \(=\) \( 2^{7} \cdot 3 \cdot 5 \)
Copy content magma:Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1920,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-368640\) \(=\) \( - 2^{13} \cdot 3^{2} \cdot 5 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8952\) \(=\)  \( 2^{3} \cdot 3 \cdot 373 \)
\( I_4 \)  \(=\) \(6072\) \(=\)  \( 2^{3} \cdot 3 \cdot 11 \cdot 23 \)
\( I_6 \)  \(=\) \(17987052\) \(=\)  \( 2^{2} \cdot 3 \cdot 1498921 \)
\( I_{10} \)  \(=\) \(1440\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 5 \)
\( J_2 \)  \(=\) \(17904\) \(=\)  \( 2^{4} \cdot 3 \cdot 373 \)
\( J_4 \)  \(=\) \(13340192\) \(=\)  \( 2^{5} \cdot 416881 \)
\( J_6 \)  \(=\) \(13237770240\) \(=\)  \( 2^{11} \cdot 3^{2} \cdot 5 \cdot 239 \cdot 601 \)
\( J_8 \)  \(=\) \(14762078945024\) \(=\)  \( 2^{8} \cdot 269 \cdot 2293 \cdot 93487 \)
\( J_{10} \)  \(=\) \(368640\) \(=\)  \( 2^{13} \cdot 3^{2} \cdot 5 \)
\( g_1 \)  \(=\) \(24952719973569408/5\)
\( g_2 \)  \(=\) \(1038436236963696/5\)
\( g_3 \)  \(=\) \(11510985848256\)

  (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

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\Q_{2}$.

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

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

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 + z^3\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + 3xz^2 + z^3\) \(0\) \(4\)

2-torsion field: \(\Q(i, \sqrt{2}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 5.004697 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.625587 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(7\) \(13\) \(2\) \(1^*\) \(1\) no
\(3\) \(1\) \(2\) \(2\) \(-1\) \(( 1 - T )( 1 + 3 T^{2} )\) yes
\(5\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) 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?
\(2\) 2.90.6 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 48.a
  Elliptic curve isogeny class 40.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 \(2\) 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 \(\Q\).

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);