Base field 4.4.2225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-5/2,1/2,1/2]),K([4,3/2,-1/2,-1/2]),K([-2,-1,1,0]),K([21,6,-11,-3]),K([-9,-57/2,-17/2,13/2])])
gp: E = ellinit([Polrev([-3,-5/2,1/2,1/2]),Polrev([4,3/2,-1/2,-1/2]),Polrev([-2,-1,1,0]),Polrev([21,6,-11,-3]),Polrev([-9,-57/2,-17/2,13/2])], K);
magma: E := EllipticCurve([K![-3,-5/2,1/2,1/2],K![4,3/2,-1/2,-1/2],K![-2,-1,1,0],K![21,6,-11,-3],K![-9,-57/2,-17/2,13/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((-a)\cdot(-1/2a^3+1/2a^2+5/2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4864a^3-4352a^2-10496a+3072)\) | = | \((-a)^{9}\cdot(-1/2a^3+1/2a^2+5/2a-1)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 18014398509481984 \) | = | \(4^{9}\cdot4^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4805055317535}{262144} a^{3} - \frac{83328303025}{32768} a^{2} + \frac{1454285809955}{16384} a + \frac{2110395388989}{32768} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(2 a^{2} - 7 a : \frac{13}{2} a^{3} - \frac{27}{2} a^{2} + \frac{5}{2} a - 6 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 11.798400570364942519153557784215456690 \) | ||
| Tamagawa product: | \( 162 \) = \(3^{2}\cdot( 2 \cdot 3^{2} )\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.12556516328474 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(4\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((-1/2a^3+1/2a^2+5/2a-1)\) | \(4\) | \(18\) | \(I_{18}\) | Split multiplicative | \(-1\) | \(1\) | \(18\) | \(18\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.