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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 221760fa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.dd3 | 221760fa1 | \([0, 0, 0, -32268, 43416592]\) | \(-19443408769/4249907200\) | \(-812169913643827200\) | \([2]\) | \(2654208\) | \(2.1158\) | \(\Gamma_0(N)\)-optimal |
| 221760.dd2 | 221760fa2 | \([0, 0, 0, -2059788, 1127734288]\) | \(5057359576472449/51765560000\) | \(9892552570306560000\) | \([2]\) | \(5308416\) | \(2.4624\) | |
| 221760.dd4 | 221760fa3 | \([0, 0, 0, 290292, -1169538032]\) | \(14156681599871/3100231750000\) | \(-592463513714688000000\) | \([2]\) | \(7962624\) | \(2.6651\) | |
| 221760.dd1 | 221760fa4 | \([0, 0, 0, -15042828, -21820184048]\) | \(1969902499564819009/63690429687500\) | \(12171430656000000000000\) | \([2]\) | \(15925248\) | \(3.0117\) |
Rank
sage: E.rank()
The elliptic curves in class 221760fa have rank \(0\).
Complex multiplication
The elliptic curves in class 221760fa do not have complex multiplication.Modular form 221760.2.a.fa
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.
