Properties

Label 44100.b
Number of curves $4$
Conductor $44100$
CM no
Rank $2$
Graph

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

Elliptic curves in class 44100.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
44100.b1 44100cp3 \([0, 0, 0, -3322200, -2152710875]\) \(189123395584/16078125\) \(344739603550781250000\) \([2]\) \(1990656\) \(2.6822\)  
44100.b2 44100cp1 \([0, 0, 0, -676200, 213474625]\) \(1594753024/4725\) \(101311230431250000\) \([2]\) \(663552\) \(2.1329\) \(\Gamma_0(N)\)-optimal
44100.b3 44100cp2 \([0, 0, 0, -400575, 389047750]\) \(-20720464/178605\) \(-61273032164820000000\) \([2]\) \(1327104\) \(2.4795\)  
44100.b4 44100cp4 \([0, 0, 0, 3568425, -9918445250]\) \(14647977776/132355125\) \(-45406342662880500000000\) \([2]\) \(3981312\) \(3.0288\)  

Rank

sage: E.rank()
 

The elliptic curves in class 44100.b have rank \(2\).

Complex multiplication

The elliptic curves in class 44100.b do not have complex multiplication.

Modular form 44100.2.a.b

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