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

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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 4 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.