Properties

Label 70644.d
Number of curves $4$
Conductor $70644$
CM no
Rank $1$
Graph

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

Elliptic curves in class 70644.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
70644.d1 70644d4 \([0, -1, 0, -1537628, -733366920]\) \(2640279346000/3087\) \(470072215533312\) \([2]\) \(870912\) \(2.0985\)  
70644.d2 70644d3 \([0, -1, 0, -95313, -11632494]\) \(-10061824000/352947\) \(-3359057706831792\) \([2]\) \(435456\) \(1.7519\)  
70644.d3 70644d2 \([0, -1, 0, -23828, -445512]\) \(9826000/5103\) \(777058152208128\) \([2]\) \(290304\) \(1.5492\)  
70644.d4 70644d1 \([0, -1, 0, 5607, -56970]\) \(2048000/1323\) \(-12591220058928\) \([2]\) \(145152\) \(1.2026\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 70644.d have rank \(1\).

Complex multiplication

The elliptic curves in class 70644.d do not have complex multiplication.

Modular form 70644.2.a.d

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