Base field \(\Q(\sqrt{85}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-21, -1, 1]))
gp: K = nfinit(Polrev([-21, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-7820,0]),K([-263580,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-7820,0]),Polrev([-263580,0])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-7820,0],K![-263580,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((11)\) | = | \((11)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 121 \) | = | \(121\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-11)\) | = | \((11)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 121 \) | = | \(121\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{52893159101157376}{11} \) | ||
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 | $\left(\frac{102975022522992971359623804098824699494562535617056170333}{167575485446229009320579984730090858963070869886833780} : -\frac{250446878911747309357386657596720693699953762954449284154212874769951350071976941849}{76695650580344479924018583491575081731491143239392335859327455423339108816561300} a + \frac{250370183261166964877462639013229118618222271811209891818353547314528010963160380549}{153391301160688959848037166983150163462982286478784671718654910846678217633122600} : 1\right)$ |
| Height | \(129.05339537948757548191956423766990605\) |
| 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: | \(1\) | ||
| Regulator: | \( 129.05339537948757548191956423766990605 \) | ||
| Period: | \( 0.064435690322791520944922286233515378905 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.8039165944255641243448666028051314497 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((11)\) | \(121\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5 and 25.
Its isogeny class
121.1-b
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 11.a1 |
| \(\Q\) | 79475.bd1 |