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Properties

Label 214774.p
Number of curves $2$
Conductor $214774$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 214774.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
214774.p1 214774f2 \([1, 1, 1, -9005707, -4702410761]\) \(545644947830040577/251340262104722\) \(37207379142165532367858\) \([2]\) \(18923520\) \(3.0256\)  
214774.p2 214774f1 \([1, 1, 1, -4556817, 3691754891]\) \(70687311717054817/1093629002564\) \(161896341630745019396\) \([2]\) \(9461760\) \(2.6791\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 214774.p have rank \(1\).

Complex multiplication

The elliptic curves in class 214774.p do not have complex multiplication.

Modular form 214774.2.a.p

sage: E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} + q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + q^{8} + q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 4 q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.