Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-2100.1-bl5
Conductor \((-20 a + 10)\)
Conductor norm \( 2100 \)
CM no
base-change yes: 1470.b6,630.h6
Q-curve yes
Torsion order \( 4 \)
Rank not available

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

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

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

Weierstrass equation

\( y^2 + a x y + \left(a + 1\right) y = x^{3} + x^{2} + \left(2887 a - 8086\right) x + 69037 a - 192755 \)
sage: E = EllipticCurve(K, [a, 1, a + 1, 2887*a - 8086, 69037*a - 192755])
 
gp: E = ellinit([a, 1, a + 1, 2887*a - 8086, 69037*a - 192755],K)
 
magma: E := ChangeRing(EllipticCurve([a, 1, a + 1, 2887*a - 8086, 69037*a - 192755]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-20 a + 10)\) = \( \left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 2100 \) = \( 3 \cdot 4 \cdot 5^{2} \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((8930250000)\) = \( \left(2\right)^{4} \cdot \left(-a + 2\right)^{12} \cdot \left(-a\right)^{6} \cdot \left(-a + 1\right)^{6} \cdot \left(a + 3\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 79749365062500000000 \) = \( 3^{12} \cdot 4^{4} \cdot 5^{12} \cdot 7^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{21302308926361}{8930250000} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(21 a - 64 : 21 a - 53 : 1\right)$,$\left(5 a - 16 : 5 a - 13 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\( \left(-a\right) \) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(a + 3\right) \) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2\right) \) \(4\) \(4\) \(I_{4}\) 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
\(3\) 3B.1.2

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.b6, 630.h6, defined over \(\Q\), so it is also a \(\Q\)-curve.