Properties

Label 2.0.4.1-67600.6-f1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 67600 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
gp: K = nfinit(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-670i+143\right){x}+3211i-5832\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,1]),K([143,-670]),K([-5832,3211])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([1,1]),Polrev([143,-670]),Polrev([-5832,3211])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,1],K![143,-670],K![-5832,3211]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((100i-240)\) = \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(2i+3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 67600 \) = \(2^{4}\cdot5\cdot5\cdot13^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((219621600i-5868800)\) = \((i+1)^{10}\cdot(-i-2)^{8}\cdot(2i+1)^{2}\cdot(2i+3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 48268090000000000 \) = \(2^{10}\cdot5^{8}\cdot5^{2}\cdot13^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{35999730234}{390625} i - \frac{51700389912}{390625} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(29 i + 2 : -95 i + 135 : 1\right)$
Height \(0.28231147803062530624508715412029056597\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-24 i - 19 : 21 i - 3 : 1\right)$ $\left(12 i + 9 : -11 i + 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.28231147803062530624508715412029056597 \)
Period: \( 0.41559372657311057118518953438641411384 \)
Tamagawa product: \( 256 \)  =  \(2^{2}\cdot2^{3}\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.7544601346915329778737150557821264372 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((i+1)\) \(2\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(4\) \(10\) \(0\)
\((-i-2)\) \(5\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((2i+1)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2i+3)\) \(13\) \(4\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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, 4 and 8.
Its isogeny class 67600.6-f consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.