# 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

# Related objects

Show commands for: Magma / SageMath

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); $\Delta$ = $-169$ = $-1 \cdot 13^{2}$

### 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: 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])

## Rational points

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);

This curve is locally solvable everywhere.

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

All rational points: (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: $0$

## 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 Tamagawa numbers: 1 (p = 13) 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