Properties

Label 441.f
Number of curves $6$
Conductor $441$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441.f1 441c5 \([1, -1, 0, -345753, -78165914]\) \(53297461115137/147\) \(12607619787\) \([2]\) \(1536\) \(1.5965\)  
441.f2 441c3 \([1, -1, 0, -21618, -1216265]\) \(13027640977/21609\) \(1853320108689\) \([2, 2]\) \(768\) \(1.2499\)  
441.f3 441c4 \([1, -1, 0, -17208, 867901]\) \(6570725617/45927\) \(3938980639167\) \([2]\) \(768\) \(1.2499\)  
441.f4 441c6 \([1, -1, 0, -15003, -1979636]\) \(-4354703137/17294403\) \(-1483273860320763\) \([2]\) \(1536\) \(1.5965\)  
441.f5 441c2 \([1, -1, 0, -1773, -5720]\) \(7189057/3969\) \(340405734249\) \([2, 2]\) \(384\) \(0.90332\)  
441.f6 441c1 \([1, -1, 0, 432, -869]\) \(103823/63\) \(-5403265623\) \([2]\) \(192\) \(0.55675\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 441.f have rank \(1\).

Complex multiplication

The elliptic curves in class 441.f do not have complex multiplication.

Modular form 441.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.