Properties

Label 2.2.21.1-2100.1-k1
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 2100 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(350a+729\right){x}-18319a-33077\)
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([1,0]),K([729,350]),K([-33077,-18319])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([729,350]),Polrev([-33077,-18319])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,-1],K![1,0],K![729,350],K![-33077,-18319]]);
 

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: \((-4033680000a-20168400000)\) = \((-a+2)^{2}\cdot(2)^{7}\cdot(-a)^{6}\cdot(-a+1)^{4}\cdot(a+3)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 406764358560000000000 \) = \(3^{2}\cdot4^{7}\cdot5^{6}\cdot5^{4}\cdot7^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{16419455525957}{100842000000} a + \frac{10311987970957}{16807000000} \)
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 $\left(-153 a + 479 : 4348 a - 12622 : 1\right)$
Height \(2.2375723040685143938326001071302379424\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{25}{4} a + \frac{27}{2} : -13 a - \frac{183}{8} : 1\right)$
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: \( 2.2375723040685143938326001071302379424 \)
Period: \( 0.39871394265609203053898939139700205022 \)
Tamagawa product: \( 32 \)  =  \(2\cdot1\cdot2\cdot2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.1149339051930901834841692784502556211 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2)\) \(4\) \(1\) \(I_{7}\) Non-split multiplicative \(1\) \(1\) \(7\) \(7\)
\((-a)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-a+1)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a+3)\) \(7\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)

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.
Its isogeny class 2100.1-k consists of curves linked by isogenies of degree 2.

Base change

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

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