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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 31680.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.e1 | 31680db4 | \([0, 0, 0, -1055753868, -13202718612208]\) | \(680995599504466943307169/52207031250000000\) | \(9976919040000000000000000\) | \([2]\) | \(13762560\) | \(3.8451\) | |
| 31680.e2 | 31680db2 | \([0, 0, 0, -70379148, -177247263472]\) | \(201738262891771037089/45727545600000000\) | \(8738670049335705600000000\) | \([2, 2]\) | \(6881280\) | \(3.4985\) | |
| 31680.e3 | 31680db1 | \([0, 0, 0, -23193228, 40619566352]\) | \(7220044159551112609/448454983680000\) | \(85701081983279431680000\) | \([2]\) | \(3440640\) | \(3.1519\) | \(\Gamma_0(N)\)-optimal |
| 31680.e4 | 31680db3 | \([0, 0, 0, 160020852, -1095253023472]\) | \(2371297246710590562911/4084000833203280000\) | \(-780464713211626420961280000\) | \([2]\) | \(13762560\) | \(3.8451\) |
Rank
sage: E.rank()
The elliptic curves in class 31680.e have rank \(0\).
Complex multiplication
The elliptic curves in class 31680.e do not have complex multiplication.Modular form 31680.2.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 LMFDB 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 LMFDB labels.
