Properties

Label 314330.d
Number of curves $2$
Conductor $314330$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 314330.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
314330.d1 314330d2 \([1, 1, 0, -12279247, 17090604581]\) \(-32391289681150609/1228250000000\) \(-7764214164934250000000\) \([]\) \(19432224\) \(2.9704\)  
314330.d2 314330d1 \([1, 1, 0, 737713, 76017829]\) \(7023836099951/4456448000\) \(-28170825716989952000\) \([]\) \(6477408\) \(2.4211\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 314330.d have rank \(1\).

Complex multiplication

The elliptic curves in class 314330.d do not have complex multiplication.

Modular form 314330.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} - q^{8} - 2 q^{9} - q^{10} - q^{12} - q^{13} + 2 q^{14} - q^{15} + q^{16} - q^{17} + 2 q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.