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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 265200.fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 265200.fk1 | 265200fk3 | \([0, 1, 0, -817408, 284175188]\) | \(1887517194957938/21849165\) | \(699173280000000\) | \([2]\) | \(2359296\) | \(1.9996\) | |
| 265200.fk2 | 265200fk2 | \([0, 1, 0, -52408, 4185188]\) | \(994958062276/98903025\) | \(1582448400000000\) | \([2, 2]\) | \(1179648\) | \(1.6530\) | |
| 265200.fk3 | 265200fk1 | \([0, 1, 0, -11908, -431812]\) | \(46689225424/7249905\) | \(28999620000000\) | \([2]\) | \(589824\) | \(1.3064\) | \(\Gamma_0(N)\)-optimal |
| 265200.fk4 | 265200fk4 | \([0, 1, 0, 64592, 20331188]\) | \(931329171502/6107473125\) | \(-195439140000000000\) | \([2]\) | \(2359296\) | \(1.9996\) |
Rank
sage: E.rank()
The elliptic curves in class 265200.fk have rank \(1\).
Complex multiplication
The elliptic curves in class 265200.fk do not have complex multiplication.Modular form 265200.2.a.fk
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.
