Properties

Label 227850.fc
Number of curves $2$
Conductor $227850$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fc1")
 
E.isogeny_class()
 

Elliptic curves in class 227850.fc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
227850.fc1 227850dk2 \([1, 1, 1, -45867088, -111556064719]\) \(5805223604235668521/435937500000000\) \(801368920898437500000000\) \([2]\) \(46448640\) \(3.3321\)  
227850.fc2 227850dk1 \([1, 1, 1, 2740912, -7729376719]\) \(1238798620042199/14760960000000\) \(-27134565360000000000000\) \([2]\) \(23224320\) \(2.9855\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 227850.fc have rank \(1\).

Complex multiplication

The elliptic curves in class 227850.fc do not have complex multiplication.

Modular form 227850.2.a.fc

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - 4 q^{11} - q^{12} - 4 q^{13} + q^{16} - 6 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.