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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 32448.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.m1 | 32448f4 | \([0, -1, 0, -2531169, 1550837313]\) | \(11339065490696/351\) | \(55515871936512\) | \([4]\) | \(516096\) | \(2.1418\) | |
| 32448.m2 | 32448f2 | \([0, -1, 0, -158409, 24203529]\) | \(22235451328/123201\) | \(2435758881214464\) | \([2, 2]\) | \(258048\) | \(1.7952\) | |
| 32448.m3 | 32448f3 | \([0, -1, 0, -70529, 50831169]\) | \(-245314376/6908733\) | \(-1092718907326365696\) | \([2]\) | \(516096\) | \(2.1418\) | |
| 32448.m4 | 32448f1 | \([0, -1, 0, -15604, -101882]\) | \(1360251712/771147\) | \(238219473915072\) | \([2]\) | \(129024\) | \(1.4486\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.m have rank \(1\).
Complex multiplication
The elliptic curves in class 32448.m do not have complex multiplication.Modular form 32448.2.a.m
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.
