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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 15680bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15680.bz3 | 15680bm1 | \([0, 0, 0, -392, -2744]\) | \(55296/5\) | \(602362880\) | \([2]\) | \(4608\) | \(0.42417\) | \(\Gamma_0(N)\)-optimal |
| 15680.bz2 | 15680bm2 | \([0, 0, 0, -1372, 16464]\) | \(148176/25\) | \(48189030400\) | \([2, 2]\) | \(9216\) | \(0.77074\) | |
| 15680.bz1 | 15680bm3 | \([0, 0, 0, -20972, 1168944]\) | \(132304644/5\) | \(38551224320\) | \([2]\) | \(18432\) | \(1.1173\) | |
| 15680.bz4 | 15680bm4 | \([0, 0, 0, 2548, 93296]\) | \(237276/625\) | \(-4818903040000\) | \([2]\) | \(18432\) | \(1.1173\) |
Rank
sage: E.rank()
The elliptic curves in class 15680bm have rank \(1\).
Complex multiplication
The elliptic curves in class 15680bm do not have complex multiplication.Modular form 15680.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
