Properties

Label 2.0.3.1-75625.1-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((275)\)
Conductor norm \( 75625 \)
CM no
Base change yes: 275.a4,2475.i4
Q-curve yes
Torsion order \( 4 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-3}) \)

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
 
gp: K = nfinit(a^2 - a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy+y=x^{3}+\left(20a-21\right)x+22\)
sage: E = EllipticCurve(K, [a + 1, 0, 1, 20*a - 21, 22])
 
gp: E = ellinit([a + 1, 0, 1, 20*a - 21, 22],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, 0, 1, 20*a - 21, 22]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((275)\) = \( \left(5\right)^{2} \cdot \left(11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 75625 \) = \( 25^{2} \cdot 121 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((859375)\) = \( \left(5\right)^{7} \cdot \left(11\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 738525390625 \) = \( 25^{7} \cdot 121 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{59319}{55} \)
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: \(2\)
Generators $\left(a - 1 : 2 a - 5 : 1\right)$ $\left(1 : -3 a - 4 : 1\right)$
Heights \(1.67487013253474\) \(1.67487013253474\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-4 a : 4 a - 15 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 2.57226317996014 \)
Period: \( 1.41649768876523 \)
Tamagawa product: \( 4 \)  =  \(2^{2}\cdot1\)
Torsion order: \(4\)
Leading coefficient: \(4.20727248113888\)
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\((j)_{-}\)
\( \left(5\right) \) \(25\) \(4\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)
\( \left(11\right) \) \(121\) \(1\) \(I_{1}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 75625.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of elliptic curves 275.a4, 2475.i4, defined over \(\Q\), so it is also a \(\Q\)-curve.