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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 4800cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4800.cc5 | 4800cb1 | \([0, 1, 0, 67, -237]\) | \(2048/3\) | \(-48000000\) | \([2]\) | \(1024\) | \(0.15937\) | \(\Gamma_0(N)\)-optimal |
4800.cc4 | 4800cb2 | \([0, 1, 0, -433, -2737]\) | \(35152/9\) | \(2304000000\) | \([2, 2]\) | \(2048\) | \(0.50594\) | |
4800.cc2 | 4800cb3 | \([0, 1, 0, -6433, -200737]\) | \(28756228/3\) | \(3072000000\) | \([2]\) | \(4096\) | \(0.85251\) | |
4800.cc3 | 4800cb4 | \([0, 1, 0, -2433, 43263]\) | \(1556068/81\) | \(82944000000\) | \([2, 2]\) | \(4096\) | \(0.85251\) | |
4800.cc1 | 4800cb5 | \([0, 1, 0, -38433, 2887263]\) | \(3065617154/9\) | \(18432000000\) | \([2]\) | \(8192\) | \(1.1991\) | |
4800.cc6 | 4800cb6 | \([0, 1, 0, 1567, 175263]\) | \(207646/6561\) | \(-13436928000000\) | \([2]\) | \(8192\) | \(1.1991\) |
Rank
sage: E.rank()
The elliptic curves in class 4800cb have rank \(1\).
Complex multiplication
The elliptic curves in class 4800cb do not have complex multiplication.Modular form 4800.2.a.cb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.