Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-525.1-k5
Conductor \((-10 a + 5)\)
Conductor norm \( 525 \)
CM no
base-change yes: 105.a1,2205.b1
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 \(1\).

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

\( y^2 + x y + y = x^{3} - 113 x - 469 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -113, -469]),K);
sage: E = EllipticCurve(K, [1, 0, 1, -113, -469])
gp (2.8): E = ellinit([1, 0, 1, -113, -469],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-10 a + 5)\) = \( \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 525 \) = \( 3 \cdot 5^{2} \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((13125)\) = \( \left(-a + 2\right)^{2} \cdot \left(-a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(a + 3\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 172265625 \) = \( 3^{2} \cdot 5^{8} \cdot 7^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{157551496201}{13125} \)
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: \(\Z/2\Z\times\Z/2\Z\)
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]
Generators: $\left(-\frac{25}{4} : \frac{21}{8} : 1\right)$,$\left(-4 a + 5 : 2 a - 3 : 1\right)$
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 \(-1\) \(1\) \(2\) \(2\)
\( \left(-a\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-a + 1\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(a + 3\right) \) \(7\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 525.1-k consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 105.a1, 2205.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.