Properties

Label 770.d
Number of curves $4$
Conductor $770$
CM no
Rank $1$
Graph

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

Elliptic curves in class 770.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.d1 770e3 \([1, -1, 0, -20909, -1158507]\) \(1010962818911303721/57392720\) \(57392720\) \([2]\) \(1024\) \(0.95539\)  
770.d2 770e4 \([1, -1, 0, -2189, 9845]\) \(1160306142246441/634128110000\) \(634128110000\) \([4]\) \(1024\) \(0.95539\)  
770.d3 770e2 \([1, -1, 0, -1309, -17787]\) \(248158561089321/1859334400\) \(1859334400\) \([2, 2]\) \(512\) \(0.60882\)  
770.d4 770e1 \([1, -1, 0, -29, -635]\) \(-2749884201/176619520\) \(-176619520\) \([2]\) \(256\) \(0.26225\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 770.d have rank \(1\).

Complex multiplication

The elliptic curves in class 770.d do not have complex multiplication.

Modular form 770.2.a.d

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