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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 121968.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.u1 | 121968bn4 | \([0, 0, 0, -41417211, 102593328266]\) | \(2970658109581346/2139291\) | \(5658267244277716992\) | \([2]\) | \(7864320\) | \(2.9102\) | |
| 121968.u2 | 121968bn3 | \([0, 0, 0, -5959371, -3325756390]\) | \(8849350367426/3314597517\) | \(8766866479784822728704\) | \([2]\) | \(7864320\) | \(2.9102\) | |
| 121968.u3 | 121968bn2 | \([0, 0, 0, -2605251, 1581321170]\) | \(1478729816932/38900169\) | \(51444041986239243264\) | \([2, 2]\) | \(3932160\) | \(2.5636\) | |
| 121968.u4 | 121968bn1 | \([0, 0, 0, 30129, 79681646]\) | \(9148592/8301447\) | \(-2744589541593892608\) | \([2]\) | \(1966080\) | \(2.2170\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.u have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.u do not have complex multiplication.Modular form 121968.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
