Base field 3.3.1129.1
Generator \(a\), with minimal polynomial \( x^{3} - 7 x - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -7, 0, 1]))
gp: K = nfinit(Polrev([-3, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([-4,-1,1]),K([-5,0,1]),K([1031,1998,-832]),K([38663,74118,-30911])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-4,-1,1]),Polrev([-5,0,1]),Polrev([1031,1998,-832]),Polrev([38663,74118,-30911])], K);
magma: E := EllipticCurve([K![1,0,0],K![-4,-1,1],K![-5,0,1],K![1031,1998,-832],K![38663,74118,-30911]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a+2)\) | = | \((a+1)\cdot(2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 24 \) | = | \(3\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((32a^2-224a-544)\) | = | \((a+1)^{7}\cdot(2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 71663616 \) | = | \(3^{7}\cdot8^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{2562158506859}{69984} a^{2} - \frac{47868240929}{2187} a - \frac{58626468463}{23328} \) | ||
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: | \(0 \le r \le 1\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(0 \le r \le 1\) | ||
Regulator: | not available | ||
Period: | \( 1.9269349556973531810159767613662545999 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.4811549784078992249438535603471480383 \) | ||
Analytic order of Ш: | not available |
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+1)\) | \(3\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
\((2)\) | \(8\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 24.2-d consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.