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SageMath
E = EllipticCurve("fw1")
E.isogeny_class()
Elliptic curves in class 232050fw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.fw4 | 232050fw1 | \([1, 0, 0, -19338, -5439708]\) | \(-51184652297689/788010612480\) | \(-12312665820000000\) | \([2]\) | \(2162688\) | \(1.7688\) | \(\Gamma_0(N)\)-optimal |
| 232050.fw3 | 232050fw2 | \([1, 0, 0, -597338, -177105708]\) | \(1508565467598193369/6280737699600\) | \(98136526556250000\) | \([2, 2]\) | \(4325376\) | \(2.1154\) | |
| 232050.fw2 | 232050fw3 | \([1, 0, 0, -894838, 17756792]\) | \(5071506329733538969/2926108608384780\) | \(45720447006012187500\) | \([2]\) | \(8650752\) | \(2.4620\) | |
| 232050.fw1 | 232050fw4 | \([1, 0, 0, -9547838, -11356280208]\) | \(6160540455434488353049/107450752500\) | \(1678918007812500\) | \([2]\) | \(8650752\) | \(2.4620\) |
Rank
sage: E.rank()
The elliptic curves in class 232050fw have rank \(1\).
Complex multiplication
The elliptic curves in class 232050fw do not have complex multiplication.Modular form 232050.2.a.fw
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.
