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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 6336cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6336.n3 | 6336cc1 | \([0, 0, 0, -3756, -87856]\) | \(30664297/297\) | \(56757583872\) | \([2]\) | \(6144\) | \(0.88325\) | \(\Gamma_0(N)\)-optimal |
6336.n2 | 6336cc2 | \([0, 0, 0, -6636, 65360]\) | \(169112377/88209\) | \(16857002409984\) | \([2, 2]\) | \(12288\) | \(1.2298\) | |
6336.n1 | 6336cc3 | \([0, 0, 0, -84396, 9427664]\) | \(347873904937/395307\) | \(75544344133632\) | \([2]\) | \(24576\) | \(1.5764\) | |
6336.n4 | 6336cc4 | \([0, 0, 0, 25044, 508880]\) | \(9090072503/5845851\) | \(-1117159523352576\) | \([2]\) | \(24576\) | \(1.5764\) |
Rank
sage: E.rank()
The elliptic curves in class 6336cc have rank \(1\).
Complex multiplication
The elliptic curves in class 6336cc do not have complex multiplication.Modular form 6336.2.a.cc
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.