Properties

Label 421344.a.842688.1
Conductor $421344$
Discriminant $842688$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 + x^2y = -7x^6 + 108x^4 - 446x^2 + 209$ (homogenize, simplify)
$y^2 + x^2zy = -7x^6 + 108x^4z^2 - 446x^2z^4 + 209z^6$ (dehomogenize, simplify)
$y^2 = -28x^6 + 433x^4 - 1784x^2 + 836$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([209, 0, -446, 0, 108, 0, -7]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![209, 0, -446, 0, 108, 0, -7], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([836, 0, -1784, 0, 433, 0, -28]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(421344\) \(=\) \( 2^{5} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(842688\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2247184\) \(=\)  \( 2^{4} \cdot 140449 \)
\( I_4 \)  \(=\) \(78904809991\) \(=\)  \( 7 \cdot 40129 \cdot 280897 \)
\( I_6 \)  \(=\) \(51716421805730562\) \(=\)  \( 2 \cdot 3^{2} \cdot 29 \cdot 99073604991821 \)
\( I_{10} \)  \(=\) \(105336\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \)
\( J_2 \)  \(=\) \(2247184\) \(=\)  \( 2^{4} \cdot 140449 \)
\( J_4 \)  \(=\) \(157806623750\) \(=\)  \( 2 \cdot 5^{4} \cdot 31 \cdot 179 \cdot 22751 \)
\( J_6 \)  \(=\) \(13134015401810688\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \cdot 401 \cdot 467 \cdot 20807 \)
\( J_8 \)  \(=\) \(1152904691832121260023\) \(=\)  \( 38014807 \cdot 30327779694689 \)
\( J_{10} \)  \(=\) \(842688\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \)
\( g_1 \)  \(=\) \(895391971763859078040408047616/13167\)
\( g_2 \)  \(=\) \(27980866301321432857668080000/13167\)
\( g_3 \)  \(=\) \(78706025099853921145856\)

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

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

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

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\Q_{3}$.

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(9533214251x^2 - 5011014400z^2\) \(=\) \(0,\) \(\)\(15\!\cdots\!40\)\(y\) \(=\) \(\)\(39\!\cdots\!19\)\(xz^2 - \)\(40\!\cdots\!00\)\(z^3\) \(48.20794\) \(\infty\)
\(D_0 - D_\infty\) \(28x^2 - 209z^2\) \(=\) \(0,\) \(56y\) \(=\) \(-209z^3\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(9533214251x^2 - 5011014400z^2\) \(=\) \(0,\) \(\)\(15\!\cdots\!40\)\(y\) \(=\) \(\)\(39\!\cdots\!19\)\(xz^2 - \)\(40\!\cdots\!00\)\(z^3\) \(48.20794\) \(\infty\)
\(D_0 - D_\infty\) \(28x^2 - 209z^2\) \(=\) \(0,\) \(56y\) \(=\) \(-209z^3\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(9533214251x^2 - 5011014400z^2\) \(=\) \(0,\) \(\)\(15\!\cdots\!40\)\(y\) \(=\) \(x^2z + \)\(79\!\cdots\!38\)\(xz^2 - \)\(81\!\cdots\!00\)\(z^3\) \(48.20794\) \(\infty\)
\(D_0 - D_\infty\) \(28x^2 - 209z^2\) \(=\) \(0,\) \(56y\) \(=\) \(x^2z - 418z^3\) \(0\) \(2\)
\(D_0 - D_\infty\) \(x^2 - 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 2xz^2 - 2z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{3}, \sqrt{1463})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(5\)
Regulator: \( 48.20794 \)
Real period: \( 0.148975 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 3.590893 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(2\) \(1 + T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 7 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 4 T + 11 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 19 T^{2} )\)

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?
\(2\) 2.180.7 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 20064.s
  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);