Properties

Base field 4.4.9909.1
Label 4.4.9909.1-21.1-d1
Conductor \((21,-a^{2} + 2 a + 3)\)
Conductor norm \( 21 \)
CM no
base-change no
Q-curve no
Torsion order \( 8 \)
Rank not available

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Base field 4.4.9909.1

Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 6*x^2 - 3*x + 3)
 
gp: K = nfinit(a^4 - 6*a^2 - 3*a + 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
 

Weierstrass equation

\( y^2 + \left(a^{2} - a - 2\right) x y + \left(a^{3} - a^{2} - 3 a + 2\right) y = x^{3} + \left(a^{3} - 5 a - 3\right) x^{2} + \left(87 a^{3} - 17 a^{2} - 418 a - 327\right) x - 740 a^{3} - 97 a^{2} + 3864 a + 3503 \)
sage: E = EllipticCurve(K, [a^2 - a - 2, a^3 - 5*a - 3, a^3 - a^2 - 3*a + 2, 87*a^3 - 17*a^2 - 418*a - 327, -740*a^3 - 97*a^2 + 3864*a + 3503])
 
gp: E = ellinit([a^2 - a - 2, a^3 - 5*a - 3, a^3 - a^2 - 3*a + 2, 87*a^3 - 17*a^2 - 418*a - 327, -740*a^3 - 97*a^2 + 3864*a + 3503],K)
 
magma: E := ChangeRing(EllipticCurve([a^2 - a - 2, a^3 - 5*a - 3, a^3 - a^2 - 3*a + 2, 87*a^3 - 17*a^2 - 418*a - 327, -740*a^3 - 97*a^2 + 3864*a + 3503]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((21,-a^{2} + 2 a + 3)\) = \( \left(-a^{3} + a^{2} + 4 a\right) \cdot \left(a^{3} - a^{2} - 4 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 21 \) = \( 3 \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((17294403,3 a + 2551848,3 a^{3} - 3 a^{2} - 12 a + 3849135,3 a^{2} - 3 a + 9819120)\) = \( \left(-a^{3} + a^{2} + 4 a\right)^{4} \cdot \left(a^{3} - a^{2} - 4 a + 1\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 466948881 \) = \( 3^{4} \cdot 7^{8} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{283422372242314}{17294403} a^{3} + \frac{566742902699713}{17294403} a^{2} - \frac{11226391012049}{5764801} a - \frac{165429125280241}{17294403} \)
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: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(-a^{3} + a^{2} + 4 a + 1 : 5 a^{3} + 6 a^{2} - 34 a - 42 : 1\right)$,$\left(-\frac{5}{2} a^{3} + \frac{5}{4} a^{2} + \frac{45}{4} a + \frac{27}{4} : -\frac{1}{2} a^{3} - \frac{7}{2} a^{2} + \frac{27}{4} a + \frac{91}{8} : 1\right)$
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\((j)_{-}\)
\( \left(-a^{3} + a^{2} + 4 a\right) \) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(a^{3} - a^{2} - 4 a + 1\right) \) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 21.1-d consists of curves linked by isogenies of degrees dividing 8.

Base change

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