Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([27,64]),K([-197,214])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([27,64]),Polrev([-197,214])], K);
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![27,64],K![-197,214]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-106a+12)\) | = | \((2)\cdot(-3a+1)\cdot(-5a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 10108 \) | = | \(4\cdot7\cdot19^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-15343616a+31617024)\) | = | \((2)^{12}\cdot(-3a+1)^{3}\cdot(-5a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 749943283253248 \) | = | \(4^{12}\cdot7^{3}\cdot19^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{646862543}{702464} a + \frac{3391232917}{1404928} \) | ||
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 | $\left(9 a - 7 : -25 a + 11 : 1\right)$ |
Height | \(0.089997492348235447653048752218161279217\) |
Torsion structure: | \(\Z/3\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-\frac{13}{3} a - \frac{5}{3} : \frac{151}{9} a - \frac{173}{9} : 1\right)$ |
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: | \( 0.089997492348235447653048752218161279217 \) | ||
Period: | \( 0.78184046481670131429407674938887217943 \) | ||
Tamagawa product: | \( 108 \) = \(( 2^{2} \cdot 3 )\cdot3\cdot3\) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.9499755349180328424676794068669113541 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
\((-3a+1)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((-5a+2)\) | \(19\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
10108.3-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.