Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-2100.1-bl11
Conductor \((-20 a + 10)\)
Conductor norm \( 2100 \)
CM no
base-change yes: 1470.b1,630.h1
Q-curve yes
Torsion order \( 12 \)
Rank not available

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)
 
gp: K = nfinit(a^2 - a - 5);
 

Weierstrass equation

\( y^2 + a x y + \left(a + 1\right) y = x^{3} + x^{2} + \left(1756162 a - 4917256\right) x - 1921823000 a + 5365074571 \)
magma: E := ChangeRing(EllipticCurve([a, 1, a + 1, 1756162*a - 4917256, -1921823000*a + 5365074571]),K);
 
sage: E = EllipticCurve(K, [a, 1, a + 1, 1756162*a - 4917256, -1921823000*a + 5365074571])
 
gp: E = ellinit([a, 1, a + 1, 1756162*a - 4917256, -1921823000*a + 5365074571],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-20 a + 10)\) = \( \left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 2100 \) = \( 3 \cdot 4 \cdot 5^{2} \cdot 7 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((41160000)\) = \( \left(2\right)^{6} \cdot \left(-a + 2\right)^{2} \cdot \left(-a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(a + 3\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 1694145600000000 \) = \( 3^{2} \cdot 4^{6} \cdot 5^{8} \cdot 7^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{4791901410190533590281}{41160000} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/6\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(-323 a + 972 : -1663 a + 4542 : 1\right)$,$\left(731 a - 1970 : 619 a - 1828 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-a\right) \) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(a + 3\right) \) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(2\right) \) \(4\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 2100.1-bl consists of curves linked by isogenies of degrees dividing 24.

Base change

This curve is the base-change of elliptic curves 1470.b1, 630.h1, defined over \(\Q\), so it is also a \(\Q\)-curve.