Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-340.1-a1
Conductor \((12 i + 14)\)
Conductor norm \( 340 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\( y^2 + \left(i + 1\right) x y + \left(i + 1\right) y = x^{3} + \left(i + 1\right) x^{2} + \left(7 i - 19\right) x - 17 i + 18 \)
magma: E := ChangeRing(EllipticCurve([i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18]),K);
 
sage: E = EllipticCurve(K, [i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18])
 
gp: E = ellinit([i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((12 i + 14)\) = \( \left(i + 1\right)^{2} \cdot \left(-i - 2\right) \cdot \left(i + 4\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 340 \) = \( 2^{2} \cdot 5 \cdot 17 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((-60884 i + 211112)\) = \( \left(i + 1\right)^{4} \cdot \left(-i - 2\right)^{3} \cdot \left(i + 4\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 48275138000 \) = \( 2^{4} \cdot 5^{3} \cdot 17^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{8285953568288}{3017196125} i + \frac{8530762181584}{3017196125} \)
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: \( 0 \)

magma: Rank(E);
 
sage: E.rank()
 

Regulator: 1

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: \(\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(-\frac{3}{2} i + 3 : -\frac{5}{4} i - \frac{11}{4} : 1\right)$
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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\( \left(-i - 2\right) \) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(i + 4\right) \) \(17\) \(2\) \(I_{6}\) 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
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 340.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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