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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 30800.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30800.cd1 | 30800bq4 | \([0, 0, 0, -8363675, 9309874250]\) | \(1010962818911303721/57392720\) | \(3673134080000000\) | \([4]\) | \(589824\) | \(2.4533\) | |
30800.cd2 | 30800bq3 | \([0, 0, 0, -875675, -74381750]\) | \(1160306142246441/634128110000\) | \(40584199040000000000\) | \([2]\) | \(589824\) | \(2.4533\) | |
30800.cd3 | 30800bq2 | \([0, 0, 0, -523675, 144914250]\) | \(248158561089321/1859334400\) | \(118997401600000000\) | \([2, 2]\) | \(294912\) | \(2.1067\) | |
30800.cd4 | 30800bq1 | \([0, 0, 0, -11675, 5138250]\) | \(-2749884201/176619520\) | \(-11303649280000000\) | \([2]\) | \(147456\) | \(1.7601\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30800.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 30800.cd do not have complex multiplication.Modular form 30800.2.a.cd
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.