Properties

Label 462.f
Number of curves $4$
Conductor $462$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 462.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462.f1 462g4 \([1, 0, 0, -14133, -647829]\) \(312196988566716625/25367712678\) \(25367712678\) \([2]\) \(576\) \(1.0416\)  
462.f2 462g3 \([1, 0, 0, -823, -11611]\) \(-61653281712625/21875235228\) \(-21875235228\) \([2]\) \(288\) \(0.69498\)  
462.f3 462g2 \([1, 0, 0, -363, 1305]\) \(5290763640625/2291573592\) \(2291573592\) \([6]\) \(192\) \(0.49225\)  
462.f4 462g1 \([1, 0, 0, 77, 161]\) \(50447927375/39517632\) \(-39517632\) \([6]\) \(96\) \(0.14568\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 462.f have rank \(0\).

Complex multiplication

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

Modular form 462.2.a.f

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.