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SageMath
E = EllipticCurve("fv1")
E.isogeny_class()
Elliptic curves in class 463680.fv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.fv1 | 463680fv4 | \([0, 0, 0, -1736443308, -76403420368]\) | \(3029968325354577848895529/1753440696000000000000\) | \(335087735245111296000000000000\) | \([2]\) | \(424673280\) | \(4.3548\) | \(\Gamma_0(N)\)-optimal* |
463680.fv2 | 463680fv2 | \([0, 0, 0, -1194533868, -15890713747792]\) | \(986396822567235411402169/6336721794060000\) | \(1210966392928925122560000\) | \([2]\) | \(141557760\) | \(3.8055\) | \(\Gamma_0(N)\)-optimal* |
463680.fv3 | 463680fv1 | \([0, 0, 0, -73223148, -258296738128]\) | \(-227196402372228188089/19338934824115200\) | \(-3695727997558451286835200\) | \([2]\) | \(70778880\) | \(3.4589\) | \(\Gamma_0(N)\)-optimal* |
463680.fv4 | 463680fv3 | \([0, 0, 0, 434109012, -9550408912]\) | \(47342661265381757089751/27397579603968000000\) | \(-5235758997515186208768000000\) | \([2]\) | \(212336640\) | \(4.0082\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.fv have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.fv do not have complex multiplication.Modular form 463680.2.a.fv
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.