Properties

Label 44400.bw
Number of curves $4$
Conductor $44400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bw1")
 
E.isogeny_class()
 

Elliptic curves in class 44400.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
44400.bw1 44400cn4 \([0, 1, 0, -4195008, 1474823988]\) \(127568139540190201/59114336463360\) \(3783317533655040000000\) \([2]\) \(3483648\) \(2.8350\)  
44400.bw2 44400cn2 \([0, 1, 0, -2125008, -1192956012]\) \(16581570075765001/998001000\) \(63872064000000000\) \([2]\) \(1161216\) \(2.2857\)  
44400.bw3 44400cn1 \([0, 1, 0, -125008, -20956012]\) \(-3375675045001/999000000\) \(-63936000000000000\) \([2]\) \(580608\) \(1.9391\) \(\Gamma_0(N)\)-optimal
44400.bw4 44400cn3 \([0, 1, 0, 924992, 174343988]\) \(1367594037332999/995878502400\) \(-63736224153600000000\) \([2]\) \(1741824\) \(2.4884\)  

Rank

sage: E.rank()
 

The elliptic curves in class 44400.bw have rank \(1\).

Complex multiplication

The elliptic curves in class 44400.bw do not have complex multiplication.

Modular form 44400.2.a.bw

sage: E.q_eigenform(10)
 
\(q + q^{3} - 4 q^{7} + q^{9} - 6 q^{11} - 2 q^{13} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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 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.