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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 121275do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.o3 | 121275do1 | \([0, 0, 1, -3675, 203656]\) | \(-4096/11\) | \(-14741052046875\) | \([]\) | \(302400\) | \(1.2143\) | \(\Gamma_0(N)\)-optimal |
| 121275.o2 | 121275do2 | \([0, 0, 1, -113925, -26807594]\) | \(-122023936/161051\) | \(-215823743018296875\) | \([]\) | \(1512000\) | \(2.0190\) | |
| 121275.o1 | 121275do3 | \([0, 0, 1, -86219175, -308144136344]\) | \(-52893159101157376/11\) | \(-14741052046875\) | \([]\) | \(7560000\) | \(2.8237\) |
Rank
sage: E.rank()
The elliptic curves in class 121275do have rank \(1\).
Complex multiplication
The elliptic curves in class 121275do do not have complex multiplication.Modular form 121275.2.a.do
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
