# Properties

 Base field $\Q(\sqrt{229})$ Label 2.2.229.1-1.1-a1 Conductor $\left(1\right)$ Conductor norm $1$ CM no base-change no Q-curve yes Torsion order $1$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $\Q(\sqrt{229})$

Generator $a$, with minimal polynomial $x^{2} - x - 57$; class number $3$.

magma: K<a> := NumberField(x^2 - x - 57);
sage: K.<a> = NumberField(x^2 - x - 57)
gp (2.8): K = nfinit(a^2 - a - 57);

## Weierstrass equation

$y^2 + a x y + a y = x^{3} + \left(22 a - 285\right) x + 307 a - 3606$
magma: E := ChangeRing(EllipticCurve([a, 0, a, 22*a - 285, 307*a - 3606]),K);
sage: E = EllipticCurve(K, [a, 0, a, 22*a - 285, 307*a - 3606])
gp (2.8): E = ellinit([a, 0, a, 22*a - 285, 307*a - 3606],K)

This is not a global minimal model: it is minimal at all primes except $\left(3, a\right)$. No global minimal model exists.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $\left(1\right)$ = $\left(1\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $1$ = $\left(1\right)$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $(\Delta)$ = $\left(146 a - 903\right)$ = $\left(3, a\right)^{12}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\Delta)$ = $531441$ = $3^{12}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $\mathfrak{D}$ = $\left(1\right)$ = $\left(1\right)$ $N(\mathfrak{D})$ = $1$ = $\left(1\right)$ $j$ = $-299510191348095 a + 2415960913292737$ 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()
Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
prime Norm Tamagawa number Kodaira symbol Reduction type ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(3, a\right)$ 3 $1$ $I_{0}$ Good 0 0 0

## Galois Representations

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

prime Image of Galois Representation
$2$ 2Cn
$3$ 3B.1.2
$5$ 5B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3, 5 and 15.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 15.

## Base change

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