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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 100800fn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.op4 | 100800fn1 | \([0, 0, 0, 33300, 3726000]\) | \(1367631/2800\) | \(-8360755200000000\) | \([2]\) | \(589824\) | \(1.7396\) | \(\Gamma_0(N)\)-optimal |
| 100800.op3 | 100800fn2 | \([0, 0, 0, -254700, 40014000]\) | \(611960049/122500\) | \(365783040000000000\) | \([2, 2]\) | \(1179648\) | \(2.0862\) | |
| 100800.op2 | 100800fn3 | \([0, 0, 0, -1262700, -510354000]\) | \(74565301329/5468750\) | \(16329600000000000000\) | \([2]\) | \(2359296\) | \(2.4327\) | |
| 100800.op1 | 100800fn4 | \([0, 0, 0, -3854700, 2912814000]\) | \(2121328796049/120050\) | \(358467379200000000\) | \([2]\) | \(2359296\) | \(2.4327\) |
Rank
sage: E.rank()
The elliptic curves in class 100800fn have rank \(1\).
Complex multiplication
The elliptic curves in class 100800fn do not have complex multiplication.Modular form 100800.2.a.fn
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.
