Properties

Label 462.a
Number of curves $4$
Conductor $462$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 462.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462.a1 462c3 \([1, 1, 0, -226, -1406]\) \(1285429208617/614922\) \(614922\) \([2]\) \(128\) \(0.065661\)  
462.a2 462c4 \([1, 1, 0, -126, 486]\) \(223980311017/4278582\) \(4278582\) \([2]\) \(128\) \(0.065661\)  
462.a3 462c2 \([1, 1, 0, -16, -20]\) \(498677257/213444\) \(213444\) \([2, 2]\) \(64\) \(-0.28091\)  
462.a4 462c1 \([1, 1, 0, 4, 0]\) \(4657463/3696\) \(-3696\) \([2]\) \(32\) \(-0.62749\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 462.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.