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SageMath
E = EllipticCurve("iu1")
E.isogeny_class()
Elliptic curves in class 227136iu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227136.dx3 | 227136iu1 | \([0, -1, 0, -4889057, -4153991583]\) | \(4649101309/6804\) | \(18914479632257384448\) | \([2]\) | \(5990400\) | \(2.6005\) | \(\Gamma_0(N)\)-optimal |
| 227136.dx4 | 227136iu2 | \([0, -1, 0, -3482977, -6594102815]\) | \(-1680914269/5786802\) | \(-16086764927234905473024\) | \([2]\) | \(11980800\) | \(2.9471\) | |
| 227136.dx1 | 227136iu3 | \([0, -1, 0, -146200097, 680212844193]\) | \(124318741396429/51631104\) | \(143529609788207344386048\) | \([2]\) | \(29952000\) | \(3.4052\) | |
| 227136.dx2 | 227136iu4 | \([0, -1, 0, -123702817, 896668173985]\) | \(-75306487574989/81352871712\) | \(-226153326722850078477778944\) | \([2]\) | \(59904000\) | \(3.7518\) |
Rank
sage: E.rank()
The elliptic curves in class 227136iu have rank \(1\).
Complex multiplication
The elliptic curves in class 227136iu do not have complex multiplication.Modular form 227136.2.a.iu
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
