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This is a model for the modular curve $X_1(16)$.

## Simplified equation

 $y^2 + y = 2x^5 - 3x^4 + x^3 + x^2 - x$ (homogenize, simplify) $y^2 + z^3y = 2x^5z - 3x^4z^2 + x^3z^3 + x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 8x^5 - 12x^4 + 4x^3 + 4x^2 - 4x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 1, -3, 2]), R());

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 1, -3, 2], R!);

sage: X = HyperellipticCurve(R([1, -4, 4, 4, -12, 8]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$256$$ $$=$$ $$2^{8}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-512$$ $$=$$ $$- 2^{9}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$26$$ $$=$$ $$2 \cdot 13$$ $$I_4$$ $$=$$ $$-2$$ $$=$$ $$-2$$ $$I_6$$ $$=$$ $$40$$ $$=$$ $$2^{3} \cdot 5$$ $$I_{10}$$ $$=$$ $$2$$ $$=$$ $$2$$ $$J_2$$ $$=$$ $$52$$ $$=$$ $$2^{2} \cdot 13$$ $$J_4$$ $$=$$ $$118$$ $$=$$ $$2 \cdot 59$$ $$J_6$$ $$=$$ $$-36$$ $$=$$ $$- 2^{2} \cdot 3^{2}$$ $$J_8$$ $$=$$ $$-3949$$ $$=$$ $$- 11 \cdot 359$$ $$J_{10}$$ $$=$$ $$512$$ $$=$$ $$2^{9}$$ $$g_1$$ $$=$$ $$742586$$ $$g_2$$ $$=$$ $$129623/4$$ $$g_3$$ $$=$$ $$-1521/8$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_4$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (1 : 0 : 2)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-4,2],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![0,-1,1],C![0,1,1],C![1,0,2],C![1,-1,1],C![1,0,0],C![1,1,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z \oplus \Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -4 : 2) - (1 : 0 : 0)$$ $$2x - z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(1 : -4 : 2) - (1 : 0 : 0)$$ $$2x - z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$2x - z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(1 : 1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$26.84182$$ Tamagawa product: $$2$$ Torsion order: $$20$$ Leading coefficient: $$0.134209$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$9$$ $$2$$ $$1 + 2 T + 2 T^{2}$$

## Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.180.3
$$3$$ 3.540.6

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_4$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{16})^+$$ with defining polynomial:
$$x^{4} - 4 x^{2} + 2$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 4.4.2048.1-1.1-a

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{16})^+$$ with defining polynomial $$x^{4} - 4 x^{2} + 2$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$4$$ 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 $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$a^{2} - 2$$ with minimal polynomial $$x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: E_2
Of $$\GL_2$$-type, simple