Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-1250.3-a2
Conductor \((25 i + 25)\)
Conductor norm \( 1250 \)
CM no
base-change yes: 400.d2,50.a2
Q-curve yes
Torsion order \( 3 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

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

Weierstrass equation

\( y^2 + x y + y = x^{3} - 76 x + 298 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -76, 298]),K);
sage: E = EllipticCurve(K, [1, 0, 1, -76, 298])
gp (2.8): E = ellinit([1, 0, 1, -76, 298],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((25 i + 25)\) = \( \left(i + 1\right) \cdot \left(-i - 2\right)^{2} \cdot \left(2 i + 1\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 1250 \) = \( 2 \cdot 5^{4} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((12500000)\) = \( \left(i + 1\right)^{10} \cdot \left(-i - 2\right)^{8} \cdot \left(2 i + 1\right)^{8} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 156250000000000 \) = \( 2^{10} \cdot 5^{16} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{121945}{32} \)
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: $\left(4 i + 5 : -16 i - 1 : 1\right)$

Height: 0.3244960492699045

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

Regulator: 0.32449604927

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

Torsion subgroup

Structure: \(\Z/3\Z\)
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]
Generator: $\left(2 : -14 : 1\right)$
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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\( \left(-i - 2\right) \) \(5\) \(3\) \(IV^*\) Additive \(-1\) \(2\) \(8\) \(0\)
\( \left(2 i + 1\right) \) \(5\) \(3\) \(IV^*\) Additive \(-1\) \(2\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1
\(5\) 5B.1.4

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 1250.3-a consists of curves linked by isogenies of degrees dividing 15.

Base change

This curve is the base-change of elliptic curves 400.d2, 50.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.