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SageMath
E = EllipticCurve("ne1")
E.isogeny_class()
Elliptic curves in class 176400.ne
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.ne1 | 176400by4 | \([0, 0, 0, -146062875, -677238493750]\) | \(502270291349/1889568\) | \(1296487341805824000000000\) | \([2]\) | \(27648000\) | \(3.4861\) | |
| 176400.ne2 | 176400by2 | \([0, 0, 0, -9352875, 11008156250]\) | \(131872229/18\) | \(12350321424000000000\) | \([2]\) | \(5529600\) | \(2.6814\) | |
| 176400.ne3 | 176400by3 | \([0, 0, 0, -4942875, -20324893750]\) | \(-19465109/248832\) | \(-170730843365376000000000\) | \([2]\) | \(13824000\) | \(3.1396\) | |
| 176400.ne4 | 176400by1 | \([0, 0, 0, -532875, 203656250]\) | \(-24389/12\) | \(-8233547616000000000\) | \([2]\) | \(2764800\) | \(2.3349\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 176400.ne have rank \(1\).
Complex multiplication
The elliptic curves in class 176400.ne do not have complex multiplication.Modular form 176400.2.a.ne
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.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
