Show commands:
SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 463680cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.cd3 | 463680cd1 | \([0, 0, 0, -73223148, 258296738128]\) | \(-227196402372228188089/19338934824115200\) | \(-3695727997558451286835200\) | \([2]\) | \(70778880\) | \(3.4589\) | \(\Gamma_0(N)\)-optimal* |
463680.cd2 | 463680cd2 | \([0, 0, 0, -1194533868, 15890713747792]\) | \(986396822567235411402169/6336721794060000\) | \(1210966392928925122560000\) | \([2]\) | \(141557760\) | \(3.8055\) | \(\Gamma_0(N)\)-optimal* |
463680.cd4 | 463680cd3 | \([0, 0, 0, 434109012, 9550408912]\) | \(47342661265381757089751/27397579603968000000\) | \(-5235758997515186208768000000\) | \([2]\) | \(212336640\) | \(4.0082\) | |
463680.cd1 | 463680cd4 | \([0, 0, 0, -1736443308, 76403420368]\) | \(3029968325354577848895529/1753440696000000000000\) | \(335087735245111296000000000000\) | \([2]\) | \(424673280\) | \(4.3548\) |
Rank
sage: E.rank()
The elliptic curves in class 463680cd have rank \(0\).
Complex multiplication
The elliptic curves in class 463680cd do not have complex multiplication.Modular form 463680.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.