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## Base field6.6.1229312.1

Generator $$a$$, with minimal polynomial $$x^{6} - 10 x^{4} + 24 x^{2} - 8$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, 0, 24, 0, -10, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-8, 0, 24, 0, -10, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}-{x}$$
sage: E = EllipticCurve([K([1,0,0,0,0,0]),K([0,0,0,0,0,0]),K([1,0,0,0,0,0]),K([-1,0,0,0,0,0]),K([0,0,0,0,0,0])])

gp: E = ellinit([Pol(Vecrev([1,0,0,0,0,0])),Pol(Vecrev([0,0,0,0,0,0])),Pol(Vecrev([1,0,0,0,0,0])),Pol(Vecrev([-1,0,0,0,0,0])),Pol(Vecrev([0,0,0,0,0,0]))], K);

magma: E := EllipticCurve([K![1,0,0,0,0,0],K![0,0,0,0,0,0],K![1,0,0,0,0,0],K![-1,0,0,0,0,0],K![0,0,0,0,0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(1/4a^5-5/2a^3+4a)$$ = $$(-1/4a^5+2a^3-1/2a^2-3a+1)\cdot(-1/2a^3+3a+1)\cdot(1/4a^5-2a^3+3a)$$ sage: E.conductor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor norm: $$392$$ = $$7\cdot7\cdot8$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E))  magma: Norm(Conductor(E)); Discriminant: $$(-28)$$ = $$(-1/4a^5+2a^3-1/2a^2-3a+1)^{3}\cdot(-1/2a^3+3a+1)^{3}\cdot(1/4a^5-2a^3+3a)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$481890304$$ = $$7^{3}\cdot7^{3}\cdot8^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{15625}{28}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/18\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{1}{2} a^{4} - 5 a^{2} + 11 : -2 a^{4} + 19 a^{2} - 40 : 1\right)$ sage: T.gens()  gp: T  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: not available sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: not available Tamagawa product: $$18$$  =  $$3\cdot3\cdot2$$ Torsion order: $$18$$ Leading coefficient: not available Analytic order of Ш: not available

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$(-1/4a^5+2a^3-1/2a^2-3a+1)$$ $$7$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(-1/2a^3+3a+1)$$ $$7$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(1/4a^5-2a^3+3a)$$ $$8$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class 392.1-k consists of curves linked by isogenies of degrees dividing 36.

## Base change

This curve is the base change of 14.a5, 448.g5, defined over $$\Q$$, so it is also a $$\Q$$-curve.