Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-75466.2-a1
Conductor \((215 i + 171)\)
Conductor norm \( 75466 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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

\( y^2 + i x y + y = x^{3} + \left(i + 1\right) x^{2} + \left(-195 i - 233\right) x + 1273 i + 1841 \)
magma: E := ChangeRing(EllipticCurve([i, i + 1, 1, -195*i - 233, 1273*i + 1841]),K);
 
sage: E = EllipticCurve(K, [i, i + 1, 1, -195*i - 233, 1273*i + 1841])
 
gp: E = ellinit([i, i + 1, 1, -195*i - 233, 1273*i + 1841],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((215 i + 171)\) = \( \left(i + 1\right) \cdot \left(-4 i + 9\right) \cdot \left(10 i + 17\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 75466 \) = \( 2 \cdot 97 \cdot 389 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((662522390 i + 1499956406)\) = \( \left(i + 1\right)^{3} \cdot \left(-4 i + 9\right) \cdot \left(10 i + 17\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 2688805137151748936 \) = \( 2^{3} \cdot 97 \cdot 389^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{2526437929797748954525}{1344402568575874468} i + \frac{265587251101817213101}{1344402568575874468} \)
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 not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

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

Torsion subgroup

Structure: Trivial
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 

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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(-4 i + 9\right) \) \(97\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(10 i + 17\right) \) \(389\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 75466.2-a consists of curves linked by isogenies of degree 3.

Base change

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