Properties

Label 2.2.21.1-2100.1-c4
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 2100 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

\({y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-95a-255\right){x}-1215a-1823\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([-255,-95]),K([-1823,-1215])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,1]),Polrev([0,0]),Polrev([-255,-95]),Polrev([-1823,-1215])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,0],K![-255,-95],K![-1823,-1215]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-20a+10)\) = \((-a+2)\cdot(2)\cdot(-a)\cdot(-a+1)\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2100 \) = \(3\cdot4\cdot5\cdot5\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((8820000a+29767500)\) = \((-a+2)^{4}\cdot(2)^{2}\cdot(-a)^{4}\cdot(-a+1)^{8}\cdot(a+3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 759691406250000 \) = \(3^{4}\cdot4^{2}\cdot5^{4}\cdot5^{8}\cdot7^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1859624173823}{229687500} a + \frac{5340875899949}{137812500} \)
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 $\left(-5 a - 1 : 3 a + 5 : 1\right)$
Height \(0.25640217623963733389829002473279487962\)
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: $\left(-\frac{5}{4} a - \frac{49}{4} : \frac{27}{4} a + \frac{25}{8} : 1\right)$ $\left(-4 a - 2 : 3 a + 10 : 1\right)$
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: \( 0.25640217623963733389829002473279487962 \)
Period: \( 2.5126937354108770560427908633011133141 \)
Tamagawa product: \( 256 \)  =  \(2\cdot2\cdot2^{2}\cdot2^{3}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 4.4988508462938427705256455777494624833 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2)\) \(4\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-a+1)\) \(5\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((a+3)\) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

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 and 4.
Its isogeny class 2100.1-c consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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