# Properties

 Label 48841.b.830297.1 Conductor $48841$ Discriminant $-830297$ Mordell-Weil group trivial Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 5x^4 + 6x^3 - 6x^2 + 2x - 1$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 5x^4z^2 + 6x^3z^3 - 6x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 8x^5 - 18x^4 + 26x^3 - 23x^2 + 10x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 2, -6, 6, -5, 2, -1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, -6, 6, -5, 2, -1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-3, 10, -23, 26, -18, 8, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$48841$$ $$=$$ $$13^{2} \cdot 17^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-830297$$ $$=$$ $$- 13^{2} \cdot 17^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$764$$ $$=$$ $$2^{2} \cdot 191$$ $$I_4$$ $$=$$ $$10777$$ $$=$$ $$13 \cdot 829$$ $$I_6$$ $$=$$ $$650195$$ $$=$$ $$5 \cdot 7 \cdot 13 \cdot 1429$$ $$I_{10}$$ $$=$$ $$106278016$$ $$=$$ $$2^{7} \cdot 13^{2} \cdot 17^{3}$$ $$J_2$$ $$=$$ $$191$$ $$=$$ $$191$$ $$J_4$$ $$=$$ $$1071$$ $$=$$ $$3^{2} \cdot 7 \cdot 17$$ $$J_6$$ $$=$$ $$30923$$ $$=$$ $$17^{2} \cdot 107$$ $$J_8$$ $$=$$ $$1189813$$ $$=$$ $$17^{2} \cdot 23 \cdot 179$$ $$J_{10}$$ $$=$$ $$830297$$ $$=$$ $$13^{2} \cdot 17^{3}$$ $$g_1$$ $$=$$ $$254194901951/830297$$ $$g_2$$ $$=$$ $$438975873/48841$$ $$g_3$$ $$=$$ $$3903467/2873$$

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

This curve has no rational points.

magma: [];

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable except over $\R$, $\Q_{2}$, and $\Q_{17}$.

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$4.401272$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$4.401272$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$13$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 13 T^{2}$$
$$17$$ $$2$$ $$3$$ $$1$$ $$1 - T + T^{2}$$

## 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
$$g_4 = \frac{15771}{16} b^{5} - \frac{4229}{2} b^{4} - \frac{40285}{16} b^{3} + \frac{54719}{8} b^{2} - \frac{7461}{4} b - \frac{3507}{4}$$
$$g_6 = \frac{6976775}{64} b^{5} - \frac{3726177}{16} b^{4} - \frac{17947943}{64} b^{3} + \frac{12075635}{16} b^{2} - \frac{6510153}{32} b - \frac{6141551}{64}$$
Conductor norm: 24137569

## 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