Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 214774.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
214774.k1 214774a2 \([1, 0, 0, -194683, 730005]\) \(5512402554625/3188422748\) \(472000996008002972\) \([2]\) \(2838528\) \(2.0808\)  
214774.k2 214774a1 \([1, 0, 0, 48657, 97321]\) \(86058173375/49827568\) \(-7376268325587952\) \([2]\) \(1419264\) \(1.7342\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 214774.k have rank \(2\).

Complex multiplication

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

Modular form 214774.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.