Properties

Label 424536.bm
Number of curves $4$
Conductor $424536$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bm1")
 
E.isogeny_class()
 

Elliptic curves in class 424536.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424536.bm1 424536bm3 \([0, -1, 0, -10486469632, 413328813423772]\) \(22501000029889239268/3620708343\) \(20521227979693303085841408\) \([2]\) \(424673280\) \(4.2593\) \(\Gamma_0(N)\)-optimal*
424536.bm2 424536bm2 \([0, -1, 0, -657399892, 6417121071460]\) \(22174957026242512/278654127129\) \(394834679335054736588595456\) \([2, 2]\) \(212336640\) \(3.9128\) \(\Gamma_0(N)\)-optimal*
424536.bm3 424536bm4 \([0, -1, 0, -112932472, 16723889332060]\) \(-28104147578308/21301741002339\) \(-120732697047666938804188867584\) \([2]\) \(424673280\) \(4.2593\)  
424536.bm4 424536bm1 \([0, -1, 0, -77112247, -102062447528]\) \(572616640141312/280535480757\) \(24843777151270658726770128\) \([2]\) \(106168320\) \(3.5662\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 424536.bm1.

Rank

sage: E.rank()
 

The elliptic curves in class 424536.bm have rank \(0\).

Complex multiplication

The elliptic curves in class 424536.bm do not have complex multiplication.

Modular form 424536.2.a.bm

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.