# Properties

 Base field $\Q(\sqrt{-2})$ Label 2.0.8.1-20898.5-a1 Conductor $(9 a + 144)$ Conductor norm $20898$ CM no base-change no Q-curve no Torsion order $1$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $\Q(\sqrt{-2})$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)
gp (2.8): K = nfinit(a^2 + 2);

## Weierstrass equation

$y^2 + x y = x^{3} - x^{2} + \left(111 a - 30\right) x + 454 a + 384$
magma: E := ChangeRing(EllipticCurve([1, -1, 0, 111*a - 30, 454*a + 384]),K);
sage: E = EllipticCurve(K, [1, -1, 0, 111*a - 30, 454*a + 384])
gp (2.8): E = ellinit([1, -1, 0, 111*a - 30, 454*a + 384],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(9 a + 144)$ = $\left(a\right) \cdot \left(-a - 1\right)^{3} \cdot \left(a - 1\right)^{2} \cdot \left(-3 a - 5\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $20898$ = $2 \cdot 3^{5} \cdot 43$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(-1062882 a - 1771470)$ = $\left(a\right)^{2} \cdot \left(-a - 1\right)^{11} \cdot \left(a - 1\right)^{11} \cdot \left(-3 a - 5\right)$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $5397542252748$ = $2^{2} \cdot 3^{22} \cdot 43$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $-\frac{2847693229}{20898} a - \frac{541607365}{20898}$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $\text{End} (E)$ = $\Z$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $\text{ST} (E)$ = $\mathrm{SU}(2)$

## 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: Trivial 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]

## 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$(j)_{-}$
$\left(a\right)$ 2 $2$ $I_{2}$ Non-split multiplicative 1 2 2
$\left(-a - 1\right)$ 3 $1$ $II^*$ Additive 3 11 0
$\left(a - 1\right)$ 3 $4$ $I_{5}^*$ Additive 2 11 5
$\left(-3 a - 5\right)$ 43 $1$ $I_{1}$ Split multiplicative 1 1 1

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 20898.5-a consists of this curve only.

## Base change

This curve is not the base-change of an elliptic curve defined over $\Q$. It is not a $\Q$-curve.