# Properties

 Label 3721.a.3721.1 Conductor 3721 Discriminant -3721 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^4 + x^3 + 3x^2 + x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 + 6x^3 + 13x^2 + 6x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 6, 13, 6, -2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$3721$$ = $$61^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-3721$$ = $$- 61^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$392$$ = $$2^{3} \cdot 7^{2}$$ $$I_4$$ = $$26596$$ = $$2^{2} \cdot 61 \cdot 109$$ $$I_6$$ = $$2436584$$ = $$2^{3} \cdot 61 \cdot 4993$$ $$I_{10}$$ = $$-15241216$$ = $$- 2^{12} \cdot 61^{2}$$ $$J_2$$ = $$49$$ = $$7^{2}$$ $$J_4$$ = $$-177$$ = $$- 3 \cdot 59$$ $$J_6$$ = $$-187$$ = $$- 11 \cdot 17$$ $$J_8$$ = $$-10123$$ = $$- 53 \cdot 191$$ $$J_{10}$$ = $$-3721$$ = $$- 61^{2}$$ $$g_1$$ = $$-282475249/3721$$ $$g_2$$ = $$20823873/3721$$ $$g_3$$ = $$448987/3721$$

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

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

## 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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(-1 : 1 : 2)$$ $$(-2 : 2 : 1)$$ $$(1 : -4 : 1)$$ $$(-1 : -4 : 2)$$ $$(-2 : 7 : 1)$$

magma: [C![-2,2,1],C![-2,7,1],C![-1,-4,2],C![-1,0,1],C![-1,1,1],C![-1,1,2],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.098760$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.098760$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.007315$$ Real period: $$28.08135$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.205419$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$61$$ $$2$$ $$2$$ $$1$$ $$1 + T + 61 T^{2}$$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ 6.6.844596301.1 with defining polynomial:
$$x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27$$

Decomposes up to isogeny as the square of the elliptic curve:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = \frac{6632447}{648} b^{5} - \frac{2121977}{162} b^{4} - \frac{325981945}{1296} b^{3} - \frac{3529355}{324} b^{2} + \frac{22472053}{18} b + \frac{130538657}{144}$$
$$g_6 = -\frac{2420355499}{324} b^{5} + \frac{25668283757}{2592} b^{4} + \frac{953942534383}{5184} b^{3} + \frac{1335649351}{2592} b^{2} - \frac{66892621411}{72} b - \frac{191864719165}{288}$$
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 $$K = \Q (a) \simeq$$ 6.6.844596301.1 with defining polynomial $$x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27$$

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 $$F \simeq$$ $$\Q(\sqrt{61})$$ with generator $$-\frac{2}{81} a^{5} + \frac{5}{81} a^{4} + \frac{29}{81} a^{3} + \frac{13}{81} a^{2} - \frac{1}{9} a - \frac{31}{9}$$ with minimal polynomial $$x^{2} - x - 15$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_3$
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 3.3.3721.1 with generator $$-\frac{4}{27} a^{5} + \frac{10}{27} a^{4} + \frac{85}{27} a^{3} - \frac{82}{27} a^{2} - \frac{44}{3} a - \frac{11}{3}$$ with minimal polynomial $$x^{3} - x^{2} - 20 x + 9$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_2$
Of $$\GL_2$$-type, simple