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SageMath
E = EllipticCurve("gw1")
E.isogeny_class()
Elliptic curves in class 270480gw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270480.gw3 | 270480gw1 | \([0, 1, 0, -172638776, 873025222740]\) | \(1180838681727016392361/692428800000\) | \(333674724930355200000\) | \([2]\) | \(35389440\) | \(3.2622\) | \(\Gamma_0(N)\)-optimal |
| 270480.gw2 | 270480gw2 | \([0, 1, 0, -173642296, 862361016404]\) | \(1201550658189465626281/28577902500000000\) | \(13771414123407360000000000\) | \([2, 2]\) | \(70778880\) | \(3.6088\) | |
| 270480.gw4 | 270480gw3 | \([0, 1, 0, 22357704, 2699351416404]\) | \(2564821295690373719/6533572090396050000\) | \(-3148465040846868014899200000\) | \([2]\) | \(141557760\) | \(3.9553\) | |
| 270480.gw1 | 270480gw4 | \([0, 1, 0, -385698616, -1657122532780]\) | \(13167998447866683762601/5158996582031250000\) | \(2486070431250000000000000000\) | \([2]\) | \(141557760\) | \(3.9553\) |
Rank
sage: E.rank()
The elliptic curves in class 270480gw have rank \(0\).
Complex multiplication
The elliptic curves in class 270480gw do not have complex multiplication.Modular form 270480.2.a.gw
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.
