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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 277200ic
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.ic4 | 277200ic1 | \([0, 0, 0, 104925, -60374750]\) | \(73929353373/954060800\) | \(-1648617062400000000\) | \([2]\) | \(3981312\) | \(2.1763\) | \(\Gamma_0(N)\)-optimal |
| 277200.ic2 | 277200ic2 | \([0, 0, 0, -1815075, -880214750]\) | \(382704614800227/27778076480\) | \(48000516157440000000\) | \([2]\) | \(7962624\) | \(2.5229\) | |
| 277200.ic3 | 277200ic3 | \([0, 0, 0, -951075, 1696457250]\) | \(-75526045083/943250000\) | \(-1188223344000000000000\) | \([2]\) | \(11943936\) | \(2.7257\) | |
| 277200.ic1 | 277200ic4 | \([0, 0, 0, -27951075, 56695457250]\) | \(1917114236485083/7117764500\) | \(8966333353824000000000\) | \([2]\) | \(23887872\) | \(3.0722\) |
Rank
sage: E.rank()
The elliptic curves in class 277200ic have rank \(1\).
Complex multiplication
The elliptic curves in class 277200ic do not have complex multiplication.Modular form 277200.2.a.ic
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.
