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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2394.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2394.b1 | 2394f4 | \([1, -1, 0, -10953, 443961]\) | \(199350693197713/547428\) | \(399075012\) | \([2]\) | \(2048\) | \(0.88357\) | |
2394.b2 | 2394f3 | \([1, -1, 0, -1953, -24111]\) | \(1130389181713/295568028\) | \(215469092412\) | \([2]\) | \(2048\) | \(0.88357\) | |
2394.b3 | 2394f2 | \([1, -1, 0, -693, 6885]\) | \(50529889873/2547216\) | \(1856920464\) | \([2, 2]\) | \(1024\) | \(0.53699\) | |
2394.b4 | 2394f1 | \([1, -1, 0, 27, 405]\) | \(2924207/102144\) | \(-74462976\) | \([2]\) | \(512\) | \(0.19042\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2394.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2394.b 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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.