Properties

Label 2.0.4.1-5525.5-b3
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 5525 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
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

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-1605i+663\right){x}-5065i+27149\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([663,-1605]),K([27149,-5065])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([663,-1605]),Polrev([27149,-5065])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![663,-1605],K![27149,-5065]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((25i-70)\) = \((-i-2)\cdot(2i+1)\cdot(-3i-2)\cdot(i+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5525 \) = \(5\cdot5\cdot13\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-19163700i-19224125)\) = \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(-3i-2)^{2}\cdot(i+4)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 736814379705625 \) = \(5^{2}\cdot5^{2}\cdot13^{2}\cdot17^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{581265201029275534788}{29472575188225} i - \frac{9608380177127950125}{1178903007529} \)
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 $\left(-21 i - \frac{25}{2} : \frac{77}{4} i + \frac{43}{4} : 1\right)$
Height \(2.2511436504131821128109078203472640521\)
Torsion structure: \(\Z/2\Z\oplus\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-20 i - 13 : 10 i + 6 : 1\right)$ $\left(-22 i - 17 : 33 i - 23 : 1\right)$
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: \( 2.2511436504131821128109078203472640521 \)
Period: \( 0.36725895727650644306344437802982913839 \)
Tamagawa product: \( 64 \)  =  \(2\cdot2\cdot2\cdot2^{3}\)
Torsion order: \(8\)
Leading coefficient: \( 1.6535053394607472108051944388094237685 \)
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\((j)_{-}\)
\((-i-2)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2i+1)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-3i-2)\) \(13\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((i+4)\) \(17\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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 5525.5-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is not a \(\Q\)-curve.

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