# Properties

 Base field 3.3.49.1 Label 3.3.49.1-49.1-a4 Conductor $(7,2 a^{2} + a - 6)$ Conductor norm $49$ CM yes ($-7$) base-change yes: 49.a4 Q-curve yes Torsion order $2$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field 3.3.49.1

Generator $a$, with minimal polynomial $x^{3} - x^{2} - 2 x + 1$; class number $1$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)
gp (2.8): K = nfinit(a^3 - a^2 - 2*a + 1);

## Weierstrass equation

$y^2 + x y = x^{3} - x^{2} - 2 x - 1$
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -2, -1]),K);
sage: E = EllipticCurve(K, [1, -1, 0, -2, -1])
gp (2.8): E = ellinit([1, -1, 0, -2, -1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(7,2 a^{2} + a - 6)$ = $\left(-a^{2} - a + 2\right)^{2}$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $49$ = $7^{2}$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(343,343 a,343 a^{2} - 686)$ = $\left(-a^{2} - a + 2\right)^{9}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $40353607$ = $7^{9}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $-3375$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $\text{End} (E)$ = $\Z[(1+\sqrt{-7})/2]$ ( Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $\text{ST} (E)$ = $N(\mathrm{U}(1))$

## Mordell-Weil rank and generators

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $\Z/2\Z$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(2 : -1 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(-a^{2} - a + 2\right)$ 7 $2$ $III^*$ Additive 2 9 0

## Galois Representations

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

prime Image of Galois Representation
$7$ 7B.1.6[3]

For all other primes $p$, the image is the normalizer of a split Cartan subgroup if $\left(\frac{ -7 }{p}\right)=+1$ or the normalizer of a nonsplit Cartan subgroup if $\left(\frac{ -7 }{p}\right)=-1$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 7 and 14.
Its isogeny class 49.1-a consists of curves linked by isogenies of degrees dividing 14.

## Base change

This curve is the base-change of elliptic curves 49.a4, defined over $\Q$, so it is also a $\Q$-curve.