Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-4,0,1]),K([-1,-1,1,0]),K([-2,0,1,0]),K([-1988,-894,238,150]),K([-8997,-50412,-7194,13305])])
gp: E = ellinit([Polrev([0,-4,0,1]),Polrev([-1,-1,1,0]),Polrev([-2,0,1,0]),Polrev([-1988,-894,238,150]),Polrev([-8997,-50412,-7194,13305])], K);
magma: E := EllipticCurve([K![0,-4,0,1],K![-1,-1,1,0],K![-2,0,1,0],K![-1988,-894,238,150],K![-8997,-50412,-7194,13305]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+2a^2+4a)\) | = | \((-a)^{4}\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 128 \) | = | \(2^{4}\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-128a^3+128a+256)\) | = | \((-a)^{14}\cdot(a^3-a^2-4a+1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -34359738368 \) | = | \(-2^{14}\cdot8^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{414403249587808247093}{128} a^{3} - \frac{696044271483193936579}{128} a^{2} - \frac{592282302218182028565}{64} a + \frac{76217710069831417535}{8} \) | ||
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: | \(1\) |
| Generator | $\left(-\frac{1872}{121} a^{3} - \frac{19}{121} a^{2} + \frac{8108}{121} a + \frac{7406}{121} : \frac{195244}{1331} a^{3} - \frac{32001}{1331} a^{2} - \frac{815779}{1331} a - \frac{510061}{1331} : 1\right)$ |
| Height | \(0.76769898660842937291335720198839658289\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(7 a^{3} - \frac{33}{4} a^{2} - \frac{141}{4} a + \frac{5}{2} : \frac{13}{2} a^{3} + \frac{67}{8} a^{2} - \frac{29}{2} a - \frac{71}{2} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.76769898660842937291335720198839658289 \) | ||
| Period: | \( 41.628049501595625370227999331574289575 \) | ||
| Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.42576698582775 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(4\) | \(I_{6}^{*}\) | Additive | \(-1\) | \(4\) | \(14\) | \(2\) |
| \((a^3-a^2-4a+1)\) | \(8\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
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 |
| \(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
128.2-a
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.