Properties

Label 169.a.169.1
Conductor 169
Discriminant -169
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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This is a model for the modular curve $X_1(13)$. The integer $13$ is the smallest $N \in \mathbb{N}$ such that $X_1(N)$ has genus $2$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 1, 1], R![1, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 1, 1]), R([1, 1, 0, 1]))

$y^2 + (x^3 + x + 1)y = x^5 + x^4$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 169 \)  =  \( 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-169\)  =  \( -1 \cdot 13^{2} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(8\)  =  \( 2^{3} \)
\( I_4 \)  =  \(3172\)  =  \( 2^{2} \cdot 13 \cdot 61 \)
\( I_6 \)  =  \(30056\)  =  \( 2^{3} \cdot 13 \cdot 17^{2} \)
\( I_{10} \)  =  \(-692224\)  =  \( -1 \cdot 2^{12} \cdot 13^{2} \)
\( J_2 \)  =  \(1\)  =  \( 1 \)
\( J_4 \)  =  \(-33\)  =  \( -1 \cdot 3 \cdot 11 \)
\( J_6 \)  =  \(-43\)  =  \( -1 \cdot 43 \)
\( J_8 \)  =  \(-283\)  =  \( -1 \cdot 283 \)
\( J_{10} \)  =  \(-169\)  =  \( -1 \cdot 13^{2} \)
\( g_1 \)  =  \(-1/169\)
\( g_2 \)  =  \(33/169\)
\( g_3 \)  =  \(43/169\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_6 \) (GAP id : [6,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_6 \) (GAP id : [12,4])
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points:

\(0\)
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

Locally solvable:

yes

Invariants of the Jacobian:

Analytic rank*:

\(0\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank:

\(0\)
magma: HasSquareSha(Jacobian(C));

Order of Ш*:

square
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion:

\(\Z/{19}\Z\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_6$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition

Splits over the number field \(\Q (b) \simeq \) 6.6.371293.1 with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 3630 b^{5} + 1720 b^{4} - 15597 b^{3} - 8353 b^{2} + 9457 b + 2644\)
\(g_6 = 316316 b^{5} + 160160 b^{4} - 1342341 b^{3} - 759759 b^{2} + 759759 b + 207207\)
Conductor norm: 1

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.371293.1 with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\) with minimal polynomial \(x^{2} - x - 3\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_3$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.169.1 with generator \(a^{3} - a^{2} - 3 a + 2\) with minimal polynomial \(x^{3} - x^{2} - 4 x - 1\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_2$
of \(\GL_2\)-type, simple