# Properties

 Label 15876.b.222264.1 Conductor $15876$ Discriminant $-222264$ Mordell-Weil group $$\Z/{3}\Z$$ Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 4x^4 + 4x^3 - 5x^2 + 2x - 1$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 4x^4z^2 + 4x^3z^3 - 5x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 8x^5 - 14x^4 + 18x^3 - 19x^2 + 10x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 2, -5, 4, -4, 2, -1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, -5, 4, -4, 2, -1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-3, 10, -19, 18, -14, 8, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$15876$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-222264$$ $$=$$ $$- 2^{3} \cdot 3^{4} \cdot 7^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$636$$ $$=$$ $$2^{2} \cdot 3 \cdot 53$$ $$I_4$$ $$=$$ $$6129$$ $$=$$ $$3^{3} \cdot 227$$ $$I_6$$ $$=$$ $$310743$$ $$=$$ $$3^{3} \cdot 17 \cdot 677$$ $$I_{10}$$ $$=$$ $$28449792$$ $$=$$ $$2^{10} \cdot 3^{4} \cdot 7^{3}$$ $$J_2$$ $$=$$ $$159$$ $$=$$ $$3 \cdot 53$$ $$J_4$$ $$=$$ $$798$$ $$=$$ $$2 \cdot 3 \cdot 7 \cdot 19$$ $$J_6$$ $$=$$ $$16268$$ $$=$$ $$2^{2} \cdot 7^{2} \cdot 83$$ $$J_8$$ $$=$$ $$487452$$ $$=$$ $$2^{2} \cdot 3 \cdot 7^{2} \cdot 829$$ $$J_{10}$$ $$=$$ $$222264$$ $$=$$ $$2^{3} \cdot 3^{4} \cdot 7^{3}$$ $$g_1$$ $$=$$ $$1254586479/2744$$ $$g_2$$ $$=$$ $$2828663/196$$ $$g_3$$ $$=$$ $$233147/126$$

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 $\R$ and $\Q_{2}$.

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: $$\Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$5.070237$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$1.690079$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$1$$ $$1 - T + T^{2}$$
$$3$$ $$4$$ $$4$$ $$1$$ $$1 + 3 T^{2}$$
$$7$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial:
$$x^{3} - 3 x - 1$$

Decomposes up to isogeny as the square of the elliptic curve:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = \frac{41553}{16} b^{2} + \frac{78489}{16} b + \frac{23085}{16}$$
$$g_6 = \frac{29032425}{64} b^{2} + \frac{54580959}{64} b + \frac{1935495}{8}$$
Conductor norm: 2744

## 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 $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

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 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$