Base field \(\Q(\sqrt{15}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 15 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 0, 1]))
gp: K = nfinit(Polrev([-15, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([128,30]),K([248,62])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([128,30]),Polrev([248,62])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![128,30],K![248,62]]);
This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((21,7a)\) | = | \((3,a)\cdot(7,a+1)\cdot(7,a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 147 \) | = | \(3\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-4032)\) | = | \((2,a+1)^{12}\cdot(3,a)^{4}\cdot(7,a+1)\cdot(7,a+6)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 16257024 \) | = | \(2^{12}\cdot3^{4}\cdot7\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
Minimal discriminant: | \((63)\) | = | \((3,a)^{4}\cdot(7,a+1)\cdot(7,a+6)\) |
Minimal discriminant norm: | \( 3969 \) | = | \(3^{4}\cdot7\cdot7\) |
j-invariant: | \( \frac{103823}{63} \) | ||
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 | $\left(\frac{5}{2} a + 8 : \frac{33}{4} a + \frac{119}{4} : 1\right)$ |
Height | \(1.4447902398017991312741211915135536128\) |
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: | $\left(-2 : a + 1 : 1\right)$ |
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: | \( 1.4447902398017991312741211915135536128 \) | ||
Period: | \( 14.607527768551696889610408346184048628 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot2^{2}\cdot1\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.4492394263207817159252064246978605005 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a+1)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((3,a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((7,a+1)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((7,a+6)\) | \(7\) | \(1\) | \(I_{1}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
147.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 63.a6 |
\(\Q\) | 8400.bn6 |