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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 316239.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 316239.n1 | 316239n6 | \([1, 1, 0, -6186539, -5925275838]\) | \(10206027697760497/5557167\) | \(14258170131123303\) | \([2]\) | \(8110080\) | \(2.4282\) | |
| 316239.n2 | 316239n4 | \([1, 1, 0, -388824, -91615005]\) | \(2533811507137/58110129\) | \(149094692605696761\) | \([2, 2]\) | \(4055040\) | \(2.0817\) | |
| 316239.n3 | 316239n2 | \([1, 1, 0, -53419, 2633800]\) | \(6570725617/2614689\) | \(6708576618621801\) | \([2, 2]\) | \(2027520\) | \(1.7351\) | |
| 316239.n4 | 316239n1 | \([1, 1, 0, -46574, 3848103]\) | \(4354703137/1617\) | \(4148779603353\) | \([2]\) | \(1013760\) | \(1.3885\) | \(\Gamma_0(N)\)-optimal |
| 316239.n5 | 316239n5 | \([1, 1, 0, 42411, -283169592]\) | \(3288008303/13504609503\) | \(-34649133245079464727\) | \([2]\) | \(8110080\) | \(2.4282\) | |
| 316239.n6 | 316239n3 | \([1, 1, 0, 172466, 19304113]\) | \(221115865823/190238433\) | \(-488099771554877097\) | \([2]\) | \(4055040\) | \(2.0817\) |
Rank
sage: E.rank()
The elliptic curves in class 316239.n have rank \(2\).
Complex multiplication
The elliptic curves in class 316239.n do not have complex multiplication.Modular form 316239.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
