Properties

Label 68952.d
Number of curves $4$
Conductor $68952$
CM no
Rank $0$
Graph

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

Elliptic curves in class 68952.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68952.d1 68952i4 \([0, -1, 0, -4974064, -4268164772]\) \(2753580869496292/39328497\) \(194387090714698752\) \([2]\) \(1720320\) \(2.4568\)  
68952.d2 68952i2 \([0, -1, 0, -319804, -62575436]\) \(2927363579728/320445801\) \(395963053127426304\) \([2, 2]\) \(860160\) \(2.1103\)  
68952.d3 68952i1 \([0, -1, 0, -75599, 6974148]\) \(618724784128/87947613\) \(6792101279310672\) \([4]\) \(430080\) \(1.7637\) \(\Gamma_0(N)\)-optimal
68952.d4 68952i3 \([0, -1, 0, 427176, -312066756]\) \(1744147297148/9513325341\) \(-47021060480887673856\) \([2]\) \(1720320\) \(2.4568\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68952.d have rank \(0\).

Complex multiplication

The elliptic curves in class 68952.d do not have complex multiplication.

Modular form 68952.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + 4 q^{7} + q^{9} + 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.