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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 277200fn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.fn4 | 277200fn1 | \([0, 0, 0, -349275, 290538250]\) | \(-100999381393/723148272\) | \(-33739205778432000000\) | \([2]\) | \(4718592\) | \(2.4303\) | \(\Gamma_0(N)\)-optimal |
| 277200.fn3 | 277200fn2 | \([0, 0, 0, -9061275, 10474866250]\) | \(1763535241378513/4612311396\) | \(215192000491776000000\) | \([2, 2]\) | \(9437184\) | \(2.7768\) | |
| 277200.fn1 | 277200fn3 | \([0, 0, 0, -144889275, 671278086250]\) | \(7209828390823479793/49509306\) | \(2309906180736000000\) | \([2]\) | \(18874368\) | \(3.1234\) | |
| 277200.fn2 | 277200fn4 | \([0, 0, 0, -12625275, 1468638250]\) | \(4770223741048753/2740574865798\) | \(127864260938671488000000\) | \([2]\) | \(18874368\) | \(3.1234\) |
Rank
sage: E.rank()
The elliptic curves in class 277200fn have rank \(0\).
Complex multiplication
The elliptic curves in class 277200fn do not have complex multiplication.Modular form 277200.2.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.
