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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 190608cl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
190608.bv4 | 190608cl1 | \([0, 1, 0, 241, 39456]\) | \(2048/891\) | \(-670686079536\) | \([2]\) | \(290304\) | \(0.94839\) | \(\Gamma_0(N)\)-optimal |
190608.bv3 | 190608cl2 | \([0, 1, 0, -16004, 754236]\) | \(37642192/1089\) | \(13115638888704\) | \([2, 2]\) | \(580608\) | \(1.2950\) | |
190608.bv1 | 190608cl3 | \([0, 1, 0, -254264, 49263972]\) | \(37736227588/33\) | \(1589774410752\) | \([2]\) | \(1161216\) | \(1.6415\) | |
190608.bv2 | 190608cl4 | \([0, 1, 0, -37664, -1749660]\) | \(122657188/43923\) | \(2115989740710912\) | \([2]\) | \(1161216\) | \(1.6415\) |
Rank
sage: E.rank()
The elliptic curves in class 190608cl have rank \(0\).
Complex multiplication
The elliptic curves in class 190608cl do not have complex multiplication.Modular form 190608.2.a.cl
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.