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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 25200fn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.fb1 | 25200fn1 | \([0, 0, 0, -630075, -192502550]\) | \(-14822892630025/42\) | \(-78382080000\) | \([]\) | \(115200\) | \(1.7470\) | \(\Gamma_0(N)\)-optimal |
| 25200.fb2 | 25200fn2 | \([0, 0, 0, 79125, -593963750]\) | \(46969655/130691232\) | \(-152438253004800000000\) | \([]\) | \(576000\) | \(2.5517\) |
Rank
sage: E.rank()
The elliptic curves in class 25200fn have rank \(1\).
Complex multiplication
The elliptic curves in class 25200fn do not have complex multiplication.Modular form 25200.2.a.fn
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
