Properties

Label 322.d
Number of curves $2$
Conductor $322$
CM no
Rank $0$
Graph

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

Elliptic curves in class 322.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
322.d1 322c2 \([1, 1, 1, -14, -23]\) \(304821217/51842\) \(51842\) \([2]\) \(48\) \(-0.37469\)  
322.d2 322c1 \([1, 1, 1, -4, 1]\) \(7189057/644\) \(644\) \([2]\) \(24\) \(-0.72126\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 322.d have rank \(0\).

Complex multiplication

The elliptic curves in class 322.d do not have complex multiplication.

Modular form 322.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} + 2q^{3} + q^{4} - 2q^{5} + 2q^{6} + q^{7} + q^{8} + q^{9} - 2q^{10} + 6q^{11} + 2q^{12} - 4q^{13} + q^{14} - 4q^{15} + q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20})\)  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.