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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2394b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2394.c2 | 2394b1 | \([1, -1, 0, -139692, 20130768]\) | \(11165451838341046875/572244736\) | \(15450607872\) | \([6]\) | \(9216\) | \(1.4280\) | \(\Gamma_0(N)\)-optimal |
| 2394.c3 | 2394b2 | \([1, -1, 0, -139452, 20203200]\) | \(-11108001800138902875/79947274872976\) | \(-2158576421570352\) | \([6]\) | \(18432\) | \(1.7746\) | |
| 2394.c1 | 2394b3 | \([1, -1, 0, -152187, 16325477]\) | \(19804628171203875/5638671302656\) | \(110985967250178048\) | \([2]\) | \(27648\) | \(1.9774\) | |
| 2394.c4 | 2394b4 | \([1, -1, 0, 400773, 107342693]\) | \(361682234074684125/462672528510976\) | \(-9106783378681540608\) | \([2]\) | \(55296\) | \(2.3239\) |
Rank
sage: E.rank()
The elliptic curves in class 2394b have rank \(1\).
Complex multiplication
The elliptic curves in class 2394b do not have complex multiplication.Modular form 2394.2.a.b
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
