Properties

Label 463680.cd
Number of curves $4$
Conductor $463680$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 463680.cd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.cd1 463680cd4 \([0, 0, 0, -1736443308, 76403420368]\) \(3029968325354577848895529/1753440696000000000000\) \(335087735245111296000000000000\) \([2]\) \(424673280\) \(4.3548\)  
463680.cd2 463680cd2 \([0, 0, 0, -1194533868, 15890713747792]\) \(986396822567235411402169/6336721794060000\) \(1210966392928925122560000\) \([2]\) \(141557760\) \(3.8055\) \(\Gamma_0(N)\)-optimal*
463680.cd3 463680cd1 \([0, 0, 0, -73223148, 258296738128]\) \(-227196402372228188089/19338934824115200\) \(-3695727997558451286835200\) \([2]\) \(70778880\) \(3.4589\) \(\Gamma_0(N)\)-optimal*
463680.cd4 463680cd3 \([0, 0, 0, 434109012, 9550408912]\) \(47342661265381757089751/27397579603968000000\) \(-5235758997515186208768000000\) \([2]\) \(212336640\) \(4.0082\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 463680.cd1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.cd have rank \(0\).

Complex multiplication

The elliptic curves in class 463680.cd do not have complex multiplication.

Modular form 463680.2.a.cd

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} + 4q^{13} + 2q^{19} + O(q^{20})\)  Toggle raw display

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.