Properties

Label 2.0.4.1-99225.2-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor \((315)\)
Conductor norm \( 99225 \)
CM no
Base change yes: 5040.v1,315.b1
Q-curve yes
Torsion order \( 3 \)
Rank \( 0 \)

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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(Pol(Vecrev([1, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{y}={x}^{3}-1182{x}-16362\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([-1182,0]),K([-16362,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-1182,0])),Pol(Vecrev([-16362,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![-1182,0],K![-16362,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((315)\) = \((-i-2)\cdot(2i+1)\cdot(3)^{2}\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 99225 \) = \(5\cdot5\cdot9^{2}\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-9966796875)\) = \((-i-2)^{9}\cdot(2i+1)^{9}\cdot(3)^{6}\cdot(7)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 99337039947509765625 \) = \(5^{9}\cdot5^{9}\cdot9^{6}\cdot49\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{250523582464}{13671875} \)
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: \(0\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-12 : -63 i : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.258325067410331 \)
Tamagawa product: \( 81 \)  =  \(3^{2}\cdot3^{2}\cdot1\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 2.32492560669298 \)
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\((j)_{-}\)
\((-i-2)\) \(5\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((2i+1)\) \(5\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((3)\) \(9\) \(1\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((7)\) \(49\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 99225.2-b consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of elliptic curves 5040.v1, 315.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.