Properties

Label 41405l
Number of curves $4$
Conductor $41405$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 41405l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41405.l3 41405l1 \([1, -1, 0, -552239, -157805712]\) \(32798729601/3185\) \(1808663567750585\) \([2]\) \(258048\) \(1.9634\) \(\Gamma_0(N)\)-optimal
41405.l2 41405l2 \([1, -1, 0, -593644, -132739125]\) \(40743095121/10144225\) \(5760593463285613225\) \([2, 2]\) \(516096\) \(2.3100\)  
41405.l4 41405l3 \([1, -1, 0, 1435201, -844052182]\) \(575722725759/874680625\) \(-496704232293504405625\) \([2]\) \(1032192\) \(2.6566\)  
41405.l1 41405l4 \([1, -1, 0, -3284969, 2182338640]\) \(6903498885921/374712065\) \(212787460082288574665\) \([2]\) \(1032192\) \(2.6566\)  

Rank

sage: E.rank()
 

The elliptic curves in class 41405l have rank \(1\).

Complex multiplication

The elliptic curves in class 41405l do not have complex multiplication.

Modular form 41405.2.a.l

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - 3 q^{8} - 3 q^{9} + q^{10} - q^{16} + 6 q^{17} - 3 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.