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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 16320.cl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16320.cl1 | 16320bn3 | \([0, 1, 0, -418625, -104390625]\) | \(30949975477232209/478125000\) | \(125337600000000\) | \([2]\) | \(147456\) | \(1.8403\) | |
16320.cl2 | 16320bn2 | \([0, 1, 0, -26945, -1535457]\) | \(8253429989329/936360000\) | \(245461155840000\) | \([2, 2]\) | \(73728\) | \(1.4937\) | |
16320.cl3 | 16320bn1 | \([0, 1, 0, -6465, 172575]\) | \(114013572049/15667200\) | \(4107062476800\) | \([2]\) | \(36864\) | \(1.1471\) | \(\Gamma_0(N)\)-optimal |
16320.cl4 | 16320bn4 | \([0, 1, 0, 37055, -7666657]\) | \(21464092074671/109596256200\) | \(-28730000985292800\) | \([4]\) | \(147456\) | \(1.8403\) |
Rank
sage: E.rank()
The elliptic curves in class 16320.cl have rank \(0\).
Complex multiplication
The elliptic curves in class 16320.cl do not have complex multiplication.Modular form 16320.2.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 LMFDB 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 LMFDB labels.