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Properties

Label 462.e
Number of curves $2$
Conductor $462$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 462.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462.e1 462e2 \([1, 1, 1, -7445, 244091]\) \(45637459887836881/13417633152\) \(13417633152\) \([2]\) \(1344\) \(0.92233\)  
462.e2 462e1 \([1, 1, 1, -405, 4731]\) \(-7347774183121/6119866368\) \(-6119866368\) \([2]\) \(672\) \(0.57576\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 462.e have rank \(1\).

Complex multiplication

The elliptic curves in class 462.e do not have complex multiplication.

Modular form 462.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.