Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-5625.2-a1
Conductor \((75)\)
Conductor norm \( 5625 \)
CM no
base-change yes: 75.a1,4800.br1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

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

\( y^2 + y = x^{3} + x^{2} - 208 x - 1256 \)
magma: E := ChangeRing(EllipticCurve([0, 1, 1, -208, -1256]),K);
sage: E = EllipticCurve(K, [0, 1, 1, -208, -1256])
gp (2.8): E = ellinit([0, 1, 1, -208, -1256],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((75)\) = \( \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 5625 \) = \( 3^{2} \cdot 25^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((29296875)\) = \( \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{10} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 858306884765625 \) = \( 3^{2} \cdot 25^{10} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{102400}{3} \)
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 group

Rank: \( 1 \)
magma: Rank(E);
sage: E.rank()

Generator: $\left(-\frac{13}{2} : \frac{33}{4} a - \frac{1}{2} : 1\right)$

Height: 2.859650869746687

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 2.85965086975

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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a - 1\right) \) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(a - 1\right) \) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(5\right) \) \(25\) \(1\) \(II^*\) Additive \(1\) \(2\) \(10\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 5625.2-a consists of curves linked by isogenies of degree5.

Base change

This curve is the base-change of elliptic curves 75.a1, 4800.br1, defined over \(\Q\), so it is also a \(\Q\)-curve.