Properties

Label 2.2.21.1-1875.1-x1
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 1875 \)
CM no
Base change yes
Q-curve yes
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

\({y}^2+{x}{y}+{y}={x}^{3}-2751{x}-104477\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-2751,0]),K([-104477,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-2751,0]),Polrev([-104477,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-2751,0],K![-104477,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-25a+50)\) = \((-a+2)\cdot(-a)^{2}\cdot(-a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1875 \) = \(3\cdot5^{2}\cdot5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3363025078125)\) = \((-a+2)^{32}\cdot(-a)^{7}\cdot(-a+1)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 11309937676097662353515625 \) = \(3^{32}\cdot5^{7}\cdot5^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{147281603041}{215233605} \)
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 $\left(\frac{81}{4} a + 167 : 405 a + \frac{15933}{8} : 1\right)$
Height \(2.6179164228440696446436609881130530853\)
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: $\left(167 : -2109 : 1\right)$
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: \( 2.6179164228440696446436609881130530853 \)
Period: \( 0.098084444109567028749682784705757222813 \)
Tamagawa product: \( 512 \)  =  \(2^{5}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.5861317358984156080273546382925390893 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(32\) \(I_{32}\) Split multiplicative \(-1\) \(1\) \(32\) \(32\)
\((-a)\) \(5\) \(4\) \(I_{1}^{*}\) Additive \(1\) \(2\) \(7\) \(1\)
\((-a+1)\) \(5\) \(4\) \(I_{1}^{*}\) Additive \(1\) \(2\) \(7\) \(1\)

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, 4, 8 and 16.
Its isogeny class 1875.1-x consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 75.b3
\(\Q\) 11025.p3