Properties

Label 2.2.21.1-1225.2-a1
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 1225 \)
CM yes (\(-3\))
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{21}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
 
gp: K = nfinit(Polrev([-5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\({y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-182a+508\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,1]),K([2,1]),K([508,-182])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,1]),Polrev([2,1]),Polrev([508,-182])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,1],K![2,1],K![508,-182]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((7a+35)\) = \((-a)^{2}\cdot(a+3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1225 \) = \(5^{2}\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2793a-29890)\) = \((-a)^{8}\cdot(a+3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 937890625 \) = \(5^{8}\cdot7^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 0 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-a + 2 : -7 a + 18 : 1\right)$
Height \(1.2485242480690012473822903854276912137\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.2485242480690012473822903854276912137 \)
Period: \( 5.0217913504766617352049737022443952993 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.7363773943612763850385046168330983840 \)
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\((j)_{-}\)
\((-a)\) \(5\) \(1\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((a+3)\) \(7\) \(1\) \(IV\) Additive \(1\) \(2\) \(4\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(7\) 7Ns.2.1

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 1225.2-a consists of curves linked by isogenies of degree 3.

Base change

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

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