Properties

Label 441.a
Number of curves $2$
Conductor $441$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 441.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441.a1 441f2 \([0, 0, 1, -8211, -286610]\) \(-1713910976512/1594323\) \(-56950811883\) \([]\) \(624\) \(0.98669\)  
441.a2 441f1 \([0, 0, 1, -21, 40]\) \(-28672/3\) \(-107163\) \([]\) \(48\) \(-0.29578\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 441.2.a.a

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

\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.