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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 21840.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.i1 | 21840b4 | \([0, -1, 0, -11889456, 15692828400]\) | \(181513839777967159549636/1202210966668359375\) | \(1231064029868400000000\) | \([2]\) | \(1228800\) | \(2.8821\) | |
| 21840.i2 | 21840b2 | \([0, -1, 0, -1212036, -101211264]\) | \(769184747004659888464/427705503364400625\) | \(109492608861286560000\) | \([2, 2]\) | \(614400\) | \(2.5356\) | |
| 21840.i3 | 21840b1 | \([0, -1, 0, -916791, -337052970]\) | \(5326172487431504287744/9384070028021325\) | \(150145120448341200\) | \([2]\) | \(307200\) | \(2.1890\) | \(\Gamma_0(N)\)-optimal |
| 21840.i4 | 21840b3 | \([0, -1, 0, 4741464, -806105664]\) | \(11512271847440983233884/6935257488834531675\) | \(-7101703668566560435200\) | \([4]\) | \(1228800\) | \(2.8821\) |
Rank
sage: E.rank()
The elliptic curves in class 21840.i have rank \(0\).
Complex multiplication
The elliptic curves in class 21840.i do not have complex multiplication.Modular form 21840.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
