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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 221760e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.mx4 | 221760e1 | \([0, 0, 0, -115212, -13040624]\) | \(885012508801/127733760\) | \(24410301671669760\) | \([2]\) | \(1179648\) | \(1.8698\) | \(\Gamma_0(N)\)-optimal |
| 221760.mx2 | 221760e2 | \([0, 0, 0, -1774092, -909499376]\) | \(3231355012744321/85377600\) | \(16315913443737600\) | \([2, 2]\) | \(2359296\) | \(2.2164\) | |
| 221760.mx3 | 221760e3 | \([0, 0, 0, -1704972, -983623664]\) | \(-2868190647517441/527295615000\) | \(-100767761258250240000\) | \([4]\) | \(4718592\) | \(2.5630\) | |
| 221760.mx1 | 221760e4 | \([0, 0, 0, -28385292, -58208735216]\) | \(13235378341603461121/9240\) | \(1765791498240\) | \([2]\) | \(4718592\) | \(2.5630\) |
Rank
sage: E.rank()
The elliptic curves in class 221760e have rank \(0\).
Complex multiplication
The elliptic curves in class 221760e do not have complex multiplication.Modular form 221760.2.a.e
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.
