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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 20160.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.ew1 | 20160v4 | \([0, 0, 0, -72252, -7474896]\) | \(129348709488/6125\) | \(1975228416000\) | \([2]\) | \(55296\) | \(1.4332\) | |
| 20160.ew2 | 20160v3 | \([0, 0, 0, -4752, -103896]\) | \(588791808/109375\) | \(2204496000000\) | \([2]\) | \(27648\) | \(1.0866\) | |
| 20160.ew3 | 20160v2 | \([0, 0, 0, -1692, 10736]\) | \(1210991472/588245\) | \(260220764160\) | \([2]\) | \(18432\) | \(0.88389\) | |
| 20160.ew4 | 20160v1 | \([0, 0, 0, -1392, 19976]\) | \(10788913152/8575\) | \(237081600\) | \([2]\) | \(9216\) | \(0.53732\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20160.ew have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.ew do not have complex multiplication.Modular form 20160.2.a.ew
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
