Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-2100.1-bl1
Conductor \((-20 a + 10)\)
Conductor norm \( 2100 \)
CM no
base-change no
Q-curve yes
Torsion order \( 6 \)
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

\( y^2 + \left(a + 1\right) x y + a y = x^{3} + \left(-a + 1\right) x^{2} + \left(-1706044 a - 3300813\right) x + 1911947903 a + 3471227548 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -a + 1, a, -1706044*a - 3300813, 1911947903*a + 3471227548]),K);
 
sage: E = EllipticCurve(K, [a + 1, -a + 1, a, -1706044*a - 3300813, 1911947903*a + 3471227548])
 
gp: E = ellinit([a + 1, -a + 1, a, -1706044*a - 3300813, 1911947903*a + 3471227548],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()
 
\(\mathfrak{D}\) = \((-359800 a + 371000)\) = \( \left(2\right)^{3} \cdot \left(-a + 2\right) \cdot \left(-a\right)^{8} \cdot \left(-a + 1\right)^{2} \cdot \left(a + 3\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 643125000000 \) = \( 3 \cdot 4^{3} \cdot 5^{10} \cdot 7^{3} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{679348670139067650666564663877375333}{229687500} a + \frac{3792515374318931303769198058884413221}{459375000} \)
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: \(\Z/6\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(315 a + 849 : 654 a + 1283 : 1\right)$
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(-a\right) \) \(5\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a + 3\right) \) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(2\right) \) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 2100.1-bl consists of curves linked by isogenies of degrees dividing 24.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.