↑

Properties

Label 17689.e.866761.1
Conductor $17689$
Discriminant $866761$
Mordell-Weil group \(\Z/{5}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^6 - x^5 + 7x^4 + 7x^3 - 28x^2 - 13x + 34$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 - x^5z + 7x^4z^2 + 7x^3z^3 - 28x^2z^4 - 13xz^5 + 34z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 4x^5 + 30x^4 + 30x^3 - 111x^2 - 50x + 137$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([34, -13, -28, 7, 7, -1, -1]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![34, -13, -28, 7, 7, -1, -1], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([137, -50, -111, 30, 30, -4, -3]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(17689\) \(=\) \( 7^{2} \cdot 19^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(866761\) \(=\) \( 7^{4} \cdot 19^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(82660\) \(=\)  \( 2^{2} \cdot 5 \cdot 4133 \)
\( I_4 \)  \(=\) \(1\) \(=\)  \( 1 \)
\( I_6 \)  \(=\) \(-22227071\) \(=\)  \( - 389 \cdot 57139 \)
\( I_{10} \)  \(=\) \(110945408\) \(=\)  \( 2^{7} \cdot 7^{4} \cdot 19^{2} \)
\( J_2 \)  \(=\) \(20665\) \(=\)  \( 5 \cdot 4133 \)
\( J_4 \)  \(=\) \(17793426\) \(=\)  \( 2 \cdot 3 \cdot 7 \cdot 41 \cdot 10333 \)
\( J_6 \)  \(=\) \(20428150568\) \(=\)  \( 2^{3} \cdot 7^{2} \cdot 83 \cdot 139 \cdot 4517 \)
\( J_8 \)  \(=\) \(26385430667561\) \(=\)  \( 7^{2} \cdot 47 \cdot 1129 \cdot 10147903 \)
\( J_{10} \)  \(=\) \(866761\) \(=\)  \( 7^{4} \cdot 19^{2} \)
\( g_1 \)  \(=\) \(3768574004844424665625/866761\)
\( g_2 \)  \(=\) \(22431988071545220750/123823\)
\( g_3 \)  \(=\) \(178034344310076200/17689\)

  (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

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 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: \(\Z/{5}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 5z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 5z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 3xz^2 - 9z^3\) \(0\) \(5\)

2-torsion field: 5.1.1132096.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 2.783151 \)
Tamagawa product: \( 5 \)
Torsion order:\( 5 \)
Leading coefficient: \( 2.226521 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number L-factor Cluster picture Tame reduction?
\(7\) \(2\) \(4\) \(5\) \(1\) \(( 1 - T )^{2}\) yes
\(19\) \(2\) \(2\) \(1\) \(1\) \(( 1 + T )^{2}\) 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?
\(2\) 2.12.2 no
\(3\) 3.72.2 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

Simple over \(\overline{\Q}\)

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);