Properties

Label 252a
Number of curves $4$
Conductor $252$
CM no
Rank $0$
Graph

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

Elliptic curves in class 252a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
252.b4 252a1 \([0, 0, 0, 60, 61]\) \(2048000/1323\) \(-15431472\) \([2]\) \(48\) \(0.068235\) \(\Gamma_0(N)\)-optimal
252.b3 252a2 \([0, 0, 0, -255, 502]\) \(9826000/5103\) \(952342272\) \([2]\) \(96\) \(0.41481\)  
252.b2 252a3 \([0, 0, 0, -1020, 12913]\) \(-10061824000/352947\) \(-4116773808\) \([6]\) \(144\) \(0.61754\)  
252.b1 252a4 \([0, 0, 0, -16455, 812446]\) \(2640279346000/3087\) \(576108288\) \([6]\) \(288\) \(0.96412\)  

Rank

sage: E.rank()
 

The elliptic curves in class 252a have rank \(0\).

Complex multiplication

The elliptic curves in class 252a do not have complex multiplication.

Modular form 252.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{7} + 6 q^{11} + 2 q^{13} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.