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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 121680.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.bc1 | 121680n4 | \([0, 0, 0, -8846643, 10124641618]\) | \(10625310339698/3855735\) | \(27785919437453998080\) | \([2]\) | \(4128768\) | \(2.6994\) | |
| 121680.bc2 | 121680n3 | \([0, 0, 0, -4587843, -3705385502]\) | \(1481943889298/34543665\) | \(248935026075287685120\) | \([2]\) | \(4128768\) | \(2.6994\) | |
| 121680.bc3 | 121680n2 | \([0, 0, 0, -633243, 109221658]\) | \(7793764996/3080025\) | \(11097926402533401600\) | \([2, 2]\) | \(2064384\) | \(2.3529\) | |
| 121680.bc4 | 121680n1 | \([0, 0, 0, 127257, 12333958]\) | \(253012016/219375\) | \(-197612649617760000\) | \([2]\) | \(1032192\) | \(2.0063\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121680.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 121680.bc do not have complex multiplication.Modular form 121680.2.a.bc
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.
