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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 139230u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.dz3 | 139230u1 | \([1, -1, 1, -91562, -10640919]\) | \(116449478628435289/1996001280\) | \(1455084933120\) | \([2]\) | \(589824\) | \(1.4632\) | \(\Gamma_0(N)\)-optimal |
| 139230.dz2 | 139230u2 | \([1, -1, 1, -94442, -9933591]\) | \(127787213284071769/15197834433600\) | \(11079221302094400\) | \([2, 2]\) | \(1179648\) | \(1.8097\) | |
| 139230.dz1 | 139230u3 | \([1, -1, 1, -369842, 76211529]\) | \(7674388308884766169/1007648705929320\) | \(734575906622474280\) | \([2]\) | \(2359296\) | \(2.1563\) | |
| 139230.dz4 | 139230u4 | \([1, -1, 1, 134878, -50844279]\) | \(372239584720800551/1745320379985000\) | \(-1272338557009065000\) | \([2]\) | \(2359296\) | \(2.1563\) |
Rank
sage: E.rank()
The elliptic curves in class 139230u have rank \(0\).
Complex multiplication
The elliptic curves in class 139230u do not have complex multiplication.Modular form 139230.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
