Properties

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

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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: 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: 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

\(\mathfrak{N} \) = \((25 i - 70)\) = \( \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(i + 4\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 5525 \) = \( 5^{2} \cdot 13 \cdot 17 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((6025625 i + 29925000)\) = \( \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(i + 4\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 931813781640625 \) = \( 5^{8} \cdot 13^{4} \cdot 17^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{226834389543384}{59636082025} i + \frac{4972600364093721}{1490902050625} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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: $\left(-5 i + \frac{41}{2} : \frac{137}{4} i - \frac{395}{4} : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 

Height: 1.1255718252065914

magma: [Height(P):P in gens];
 
sage: [P.height() for P in gens]
 

Regulator: 1.12557182521

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

Torsion subgroup

Structure: \(\Z/4\Z\times\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
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: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(-i - 2\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2 i + 1\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-3 i - 2\right) \) \(13\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(i + 4\right) \) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 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.