Properties

Label 405600ef
Number of curves $2$
Conductor $405600$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("ef1")
 
E.isogeny_class()
 

Elliptic curves in class 405600ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.ef2 405600ef1 \([0, 1, 0, 282, 16368]\) \(64/3\) \(-115843416000\) \([2]\) \(442368\) \(0.80272\) \(\Gamma_0(N)\)-optimal*
405600.ef1 405600ef2 \([0, 1, 0, -8168, 269868]\) \(195112/9\) \(2780241984000\) \([2]\) \(884736\) \(1.1493\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 405600ef1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600ef have rank \(1\).

Complex multiplication

The elliptic curves in class 405600ef do not have complex multiplication.

Modular form 405600.2.a.ef

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{7} + q^{9} - 6 q^{11} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.