# Properties

 Label 2.2.152.1-38.1-b3 Base field $$\Q(\sqrt{38})$$ Conductor norm $$38$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$2$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{38})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 38$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-38, 0, 1]))

gp: K = nfinit(Polrev([-38, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-38, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-4201a+25979\right){x}+2939458a-18119833$$
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,1]),K([25979,-4201]),K([-18119833,2939458])])

gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,1]),Polrev([25979,-4201]),Polrev([-18119833,2939458])], K);

magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,1],K![25979,-4201],K![-18119833,2939458]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a)$$ = $$(-a-6)\cdot(3a-19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$38$$ = $$2\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3511808)$$ = $$(-a-6)^{18}\cdot(3a-19)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$12332795428864$$ = $$2^{18}\cdot19^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{94196375}{3511808}$$ 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: $$2$$ Generators $\left(-\frac{55136670}{259081} a + \frac{338344399}{259081} : \frac{1364354240035}{131872229} a - \frac{8408107536066}{131872229} : 1\right)$ $\left(-342 a + 2103 : -23073 a + 142246 : 1\right)$ Heights $$5.3534403960400350403975627472602030892$$ $$0.56613502312489363059399202396476133602$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.14619032273225034088597563525462278176$$ Period: $$1.4463135242216651559100740810240158370$$ Tamagawa product: $$108$$  =  $$( 2 \cdot 3^{2} )\cdot( 2 \cdot 3 )$$ Torsion order: $$1$$ Leading coefficient: $$7.4087173261964803561355291676385075641$$ 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$$(j)_{-}$$
$$(-a-6)$$ $$2$$ $$18$$ $$I_{18}$$ Split multiplicative $$-1$$ $$1$$ $$18$$ $$18$$
$$(3a-19)$$ $$19$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 38.1-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 722.e3
$$\Q$$ 1216.m3