Properties

 Label 2.0.7.1-28672.7-i4 Base field $$\Q(\sqrt{-7})$$ Conductor norm $$28672$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-7})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 2$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))

gp: K = nfinit(Pol(Vecrev([2, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);

Weierstrass equation

$${y}^2={x}^{3}-a{x}^{2}+\left(43a+126\right){x}+439a-618$$
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([126,43]),K([-618,439])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([126,43])),Pol(Vecrev([-618,439]))], K);

magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![126,43],K![-618,439]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(-128a+64)$$ = $$(a)^{6}\cdot(-a+1)^{6}\cdot(-2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$28672$$ = $$2^{6}\cdot2^{6}\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-32768a-98304)$$ = $$(a)^{15}\cdot(-a+1)^{16}\cdot(-2a+1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$15032385536$$ = $$2^{15}\cdot2^{16}\cdot7$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{12673028}{7} a + \frac{12334320}{7}$$ 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: $$1$$ Generator $\left(-5 a + 8 : -8 a + 20 : 1\right)$ Height $$1.33348909857216$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-5 a + 4 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.33348909857216$$ Period: $$1.08753612818808$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$4.38504576050617$$ Analytic order of Ш: $$1$$ (rounded)

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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{5}^{*}$$ Additive $$1$$ $$6$$ $$15$$ $$0$$
$$(-a+1)$$ $$2$$ $$4$$ $$I_{6}^{*}$$ Additive $$1$$ $$6$$ $$16$$ $$0$$
$$(-2a+1)$$ $$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 28672.7-i consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.