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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

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

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: X = HyperellipticCurve(R([1, 2, 1, 2, 6, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

 $$I_2$$ $$=$$ $$4$$ $$=$$ $$2^{2}$$ $$I_4$$ $$=$$ $$793$$ $$=$$ $$13 \cdot 61$$ $$I_6$$ $$=$$ $$3757$$ $$=$$ $$13 \cdot 17^{2}$$ $$I_{10}$$ $$=$$ $$-21632$$ $$=$$ $$- 2^{7} \cdot 13^{2}$$ $$J_2$$ $$=$$ $$1$$ $$=$$ $$1$$ $$J_4$$ $$=$$ $$-33$$ $$=$$ $$- 3 \cdot 11$$ $$J_6$$ $$=$$ $$-43$$ $$=$$ $$-43$$ $$J_8$$ $$=$$ $$-283$$ $$=$$ $$-283$$ $$J_{10}$$ $$=$$ $$-169$$ $$=$$ $$- 13^{2}$$ $$g_1$$ $$=$$ $$-1/169$$ $$g_2$$ $$=$$ $$33/169$$ $$g_3$$ $$=$$ $$43/169$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

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

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (-1 : 1 : 1)$$

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]];

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable everywhere.

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$19$$

## BSD invariants

 Hasse-Weil conjecture: verified 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

## 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

The conductor $169$ of the Jacobian of $X_1(13)$ is the smallest known to arise for an abelian surface (and for any rational $L$-function of motivic weight $1$ and degree $4$).
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 $13$.