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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 $2$.

## Simplified 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]))

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([1, 2, 1, 2, 6, 4, 1]))

 $y^2 + (x^3 + x + 1)y = x^5 + x^4$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^5z + x^4z^2$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 6x^4 + 2x^3 + x^2 + 2x + 1$ (minimize, homogenize)

## Invariants

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

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$8$$ = $$2^{3}$$ $$I_4$$ = $$3172$$ = $$2^{2} \cdot 13 \cdot 61$$ $$I_6$$ = $$30056$$ = $$2^{3} \cdot 13 \cdot 17^{2}$$ $$I_{10}$$ = $$-692224$$ = $$- 2^{12} \cdot 13^{2}$$ $$J_2$$ = $$1$$ = $$1$$ $$J_4$$ = $$-33$$ = $$- 3 \cdot 11$$ $$J_6$$ = $$-43$$ = $$- 43$$ $$J_8$$ = $$-283$$ = $$- 283$$ $$J_{10}$$ = $$-169$$ = $$- 13^{2}$$ $$g_1$$ = $$-1/169$$ $$g_2$$ = $$33/169$$ $$g_3$$ = $$43/169$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_6$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $D_6$

## Rational points

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]];

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 0),\, (-1 : 1 : 1)$$

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

Number of rational Weierstrass points: $$0$$

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.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{19}\Z$$

Generator Height Order
$$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$19$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$32.66703$$ Tamagawa product: $$1$$ Torsion order: $$19$$ Leading coefficient: $$0.090490$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$13$$ $$2$$ $$2$$ $$1$$ $$1 + 5 T + 13 T^{2}$$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{13})^+$$ 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 the Jacobian

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 $$K = \Q (a) \simeq$$ $$\Q(\zeta_{13})^+$$ 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

Mazur and Tate (Invent. Math. 22 (1973), 41-49) attribute the discovery of $19$-torsion point on the Jacobian of $X_1(13)$ to Ogg, and use it to prove that this point generates the full Mordell-Weil group using a 19-descent(!). It follows that there are no rational points other than the three pairs above $x=0,-1,\infty$; since these points are cusps of $X_1(13)$, Mazur and Tate thus prove that there is no elliptic curve $E/\Q$ with a rational torsion point of order $13$.