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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 33635.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33635.e1 | 33635k4 | \([1, -1, 1, -1274947, 553996574]\) | \(258243633650241/226262645\) | \(200808930310296245\) | \([2]\) | \(368640\) | \(2.2458\) | |
| 33635.e2 | 33635k3 | \([1, -1, 1, -842497, -294335786]\) | \(74517479217441/893544155\) | \(793023726698534555\) | \([2]\) | \(368640\) | \(2.2458\) | |
| 33635.e3 | 33635k2 | \([1, -1, 1, -97722, 4467944]\) | \(116285729841/57684025\) | \(51194784522396025\) | \([2, 2]\) | \(184320\) | \(1.8992\) | |
| 33635.e4 | 33635k1 | \([1, -1, 1, 22403, 527844]\) | \(1401168159/949375\) | \(-842573807149375\) | \([4]\) | \(92160\) | \(1.5526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33635.e have rank \(1\).
Complex multiplication
The elliptic curves in class 33635.e do not have complex multiplication.Modular form 33635.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
