# Properties

 Label 2.0.7.1-23716.5-n2 Base field $$\Q(\sqrt{-7})$$ Conductor norm $$23716$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([2, -1, 1]));

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

## Weierstrass equation

$${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1044a-1498\right){x}+19529a-9470$$
sage: E = EllipticCurve([K([1,0]),K([0,-1]),K([0,1]),K([-1498,1044]),K([-9470,19529])])

gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([-1498,1044]),Polrev([-9470,19529])], K);

magma: E := EllipticCurve([K![1,0],K![0,-1],K![0,1],K![-1498,1044],K![-9470,19529]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(154)$$ = $$(a)\cdot(-a+1)\cdot(-2a+1)^{2}\cdot(-2a+3)\cdot(2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$23716$$ = $$2\cdot2\cdot7^{2}\cdot11\cdot11$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-56512151696a+11526097968)$$ = $$(a)^{4}\cdot(-a+1)^{8}\cdot(-2a+1)^{3}\cdot(-2a+3)^{5}\cdot(2a+1)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$5868732916160791728128$$ = $$2^{4}\cdot2^{8}\cdot7^{3}\cdot11^{5}\cdot11^{10}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{30861084894361475}{6639980697856} a + \frac{18691001478000609}{3319990348928}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{33}{4} a - \frac{127}{4} : -\frac{37}{8} a + \frac{127}{8} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.19560816973530126264541339297633292764$$ Tamagawa product: $$128$$  =  $$2^{2}\cdot2^{3}\cdot2\cdot1\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$4.7317080825793671845869824408308366860$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(-a+1)$$ $$2$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$(-2a+1)$$ $$7$$ $$2$$ $$III$$ Additive $$-1$$ $$2$$ $$3$$ $$0$$
$$(-2a+3)$$ $$11$$ $$1$$ $$I_{5}$$ Non-split multiplicative $$1$$ $$1$$ $$5$$ $$5$$
$$(2a+1)$$ $$11$$ $$2$$ $$I_{10}$$ Non-split multiplicative $$1$$ $$1$$ $$10$$ $$10$$

## 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.
Its isogeny class 23716.5-n consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is a $$\Q$$-curve.

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