Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-5525.5-b9
Conductor \( \left(25 i - 70\right) \)
Conductor norm \( 5525 \)
CM no
base-change no
Q-curve no
Torsion order \( 16 \)
Rank \( 1 \)

Related objects

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Show commands for: Magma / SageMath / Pari/GP

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

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

magma: K<i> := NumberField(x^2 + 1);
sage: K.<i> = NumberField(x^2 + 1)
gp (2.8): K = nfinit(i^2 + 1);

Weierstrass equation

\( y^2 + i x y + i y = x^{3} + x^{2} + \left(-105 i + 39\right) x + 15 i - 399 \)
magma: E := ChangeRing(EllipticCurve([i, 1, i, -105*i + 39, 15*i - 399]),K);
sage: E = EllipticCurve(K, [i, 1, i, -105*i + 39, 15*i - 399])
gp (2.8): E = ellinit([i, 1, i, -105*i + 39, 15*i - 399],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \( \left(25 i - 70\right) \) = \( \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(i + 4\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 5525 \) = \( 5^{2} \cdot 13 \cdot 17 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \( \left(6025625 i + 29925000\right) \) = \( \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(i + 4\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 931813781640625 \) = \( 5^{8} \cdot 13^{4} \cdot 17^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{226834389543384}{59636082025} i + \frac{4972600364093721}{1490902050625} \)
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: \( 1 \)
magma: Rank(E);
sage: E.rank()

Generator: $\left(-5 i + \frac{41}{2} : \frac{137}{4} i - \frac{395}{4} : 1\right)$

Height: 1.12557182521

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

Regulator: 1.12557182521

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

Torsion subgroup

Structure: \(\Z/4\Z\times\Z/4\Z\)
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]
Generators: $\left(-25 i - 17 : 23 i - 165 : 1\right)$,$\left(10 i - \frac{9}{2} : -\frac{13}{4} i - \frac{145}{4} : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-i - 2\right) \) 5 \(4\) \( I_{4} \) Split multiplicative 1 4 4
\( \left(2 i + 1\right) \) 5 \(4\) \( I_{4} \) Split multiplicative 1 4 4
\( \left(-3 i - 2\right) \) 13 \(4\) \( I_{4} \) Split multiplicative 1 4 4
\( \left(i + 4\right) \) 17 \(4\) \( I_{4} \) Split multiplicative 1 4 4

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 5525.5-b consists of 10 curves linked by isogenies of degrees dividing 16.

Base change

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