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([2,-2,-1,1]),K([-4,1,2,-1]),K([-2,-1,1,0]),K([45,-11,-90,33]),K([327,-45,-606,247])])
gp: E = ellinit([Polrev([2,-2,-1,1]),Polrev([-4,1,2,-1]),Polrev([-2,-1,1,0]),Polrev([45,-11,-90,33]),Polrev([327,-45,-606,247])], K);
magma: E := EllipticCurve([K![2,-2,-1,1],K![-4,1,2,-1],K![-2,-1,1,0],K![45,-11,-90,33],K![327,-45,-606,247]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^2+8)\) | = | \((-a)^{6}\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 512 \) | = | \(2^{6}\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-62a^3-86a^2+140a+352)\) | = | \((-a)^{30}\cdot(a^3-a^2-4a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -8589934592 \) | = | \(-2^{30}\cdot8\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{14270157697541427}{4096} a^{3} + \frac{35796422109839569}{4096} a^{2} + \frac{1541200524057185}{2048} a - \frac{18919901006348423}{4096} \) | ||
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 | $\left(\frac{1}{4} a^{3} - \frac{11}{4} a^{2} + \frac{11}{2} a + \frac{21}{4} : -\frac{35}{8} a^{3} - \frac{23}{8} a^{2} + \frac{61}{4} a + \frac{81}{8} : 1\right)$ |
| Height | \(0.62510878683097468281439439059880934540\) |
| 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: | $\left(\frac{1}{4} a^{3} - \frac{15}{4} a^{2} + \frac{5}{2} a + 3 : \frac{15}{8} a^{3} - \frac{3}{8} a^{2} - \frac{5}{4} a + 1 : 1\right)$ |
| 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: | \( 0.62510878683097468281439439059880934540 \) | ||
| Period: | \( 73.431827893428427288239874489831520574 \) | ||
| Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.48427154369305 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(4\) | \(I_{20}^{*}\) | Additive | \(-1\) | \(6\) | \(30\) | \(12\) |
| \((a^3-a^2-4a+1)\) | \(8\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 |
| \(3\) | 3B |
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
512.3-l
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.