Properties

Label 405600.b
Number of curves $4$
Conductor $405600$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 405600.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.b1 405600b2 \([0, -1, 0, -136608, 19472712]\) \(7301384/3\) \(115843416000000\) \([2]\) \(2359296\) \(1.6610\) \(\Gamma_0(N)\)-optimal*
405600.b2 405600b4 \([0, -1, 0, -73233, -7461663]\) \(140608/3\) \(926747328000000\) \([2]\) \(2359296\) \(1.6610\)  
405600.b3 405600b1 \([0, -1, 0, -9858, 206712]\) \(21952/9\) \(43441281000000\) \([2, 2]\) \(1179648\) \(1.3144\) \(\Gamma_0(N)\)-optimal*
405600.b4 405600b3 \([0, -1, 0, 32392, 1474212]\) \(97336/81\) \(-3127772232000000\) \([2]\) \(2359296\) \(1.6610\)  
*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 405600.b1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 405600.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.