Show commands:
SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 162240.eh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.eh1 | 162240z3 | \([0, 1, 0, -1119681, 453929919]\) | \(490757540836/2142075\) | \(677602059136204800\) | \([2]\) | \(4128768\) | \(2.2744\) | |
| 162240.eh2 | 162240z2 | \([0, 1, 0, -105681, -950481]\) | \(1650587344/950625\) | \(75177743247360000\) | \([2, 2]\) | \(2064384\) | \(1.9278\) | |
| 162240.eh3 | 162240z1 | \([0, 1, 0, -75261, -7953165]\) | \(9538484224/26325\) | \(130115324851200\) | \([2]\) | \(1032192\) | \(1.5812\) | \(\Gamma_0(N)\)-optimal |
| 162240.eh4 | 162240z4 | \([0, 1, 0, 421599, -7172385]\) | \(26198797244/15234375\) | \(-4819086105600000000\) | \([2]\) | \(4128768\) | \(2.2744\) |
Rank
sage: E.rank()
The elliptic curves in class 162240.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 162240.eh do not have complex multiplication.Modular form 162240.2.a.eh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
