Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-300.1-a3
Conductor \((-10 a + 20)\)
Conductor norm \( 300 \)
CM no
base-change yes: 30.a2,4410.z2
Q-curve yes
Torsion order \( 2 \)
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 (2.8): K = nfinit(a^2 - a - 5);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} - 454 x - 544 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -454, -544]),K);
 
sage: E = EllipticCurve(K, [1, 0, 1, -454, -544])
 
gp (2.8): E = ellinit([1, 0, 1, -454, -544],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-10 a + 20)\) = \( \left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 300 \) = \( 3 \cdot 4 \cdot 5^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((5859375000)\) = \( \left(2\right)^{3} \cdot \left(-a + 2\right)^{2} \cdot \left(-a\right)^{12} \cdot \left(-a + 1\right)^{12} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 34332275390625000000 \) = \( 3^{2} \cdot 4^{3} \cdot 5^{24} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{10316097499609}{5859375000} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): 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()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(\frac{87}{4} : -\frac{91}{8} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(-a\right) \) \(5\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\( \left(2\right) \) \(4\) \(3\) \(I_{3}\) 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.1.2

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 300.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 30.a2, 4410.z2, defined over \(\Q\), so it is also a \(\Q\)-curve.