Properties

Label 61710.da
Number of curves $8$
Conductor $61710$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("da1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 61710.da

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.da1 61710cz8 \([1, 0, 0, -24349614385, 1462465543989737]\) \(901247067798311192691198986281/552431869440\) \(978666755056995840\) \([2]\) \(79626240\) \(4.1597\)  
61710.da2 61710cz7 \([1, 0, 0, -1532073265, 22528378613225]\) \(224494757451893010998773801/6152490825146276160000\) \(10899512798686962140285760000\) \([2]\) \(79626240\) \(4.1597\)  
61710.da3 61710cz6 \([1, 0, 0, -1521851185, 22850919992297]\) \(220031146443748723000125481/172266701724057600\) \(305180970372973205913600\) \([2, 2]\) \(39813120\) \(3.8132\)  
61710.da4 61710cz5 \([1, 0, 0, -300673810, 2005244394572]\) \(1696892787277117093383481/1440538624914939000\) \(2552002046892934249779000\) \([2]\) \(26542080\) \(3.6104\)  
61710.da5 61710cz4 \([1, 0, 0, -196913890, -1052215504900]\) \(476646772170172569823801/5862293314453125000\) \(10385410206445892578125000\) \([2]\) \(26542080\) \(3.6104\)  
61710.da6 61710cz3 \([1, 0, 0, -94477105, 362070412265]\) \(-52643812360427830814761/1504091705903677440\) \(-2664590206602424709283840\) \([4]\) \(19906560\) \(3.4666\)  
61710.da7 61710cz2 \([1, 0, 0, -22978810, 16337265572]\) \(757443433548897303481/373234243041000000\) \(661207228835957001000000\) \([2, 2]\) \(13271040\) \(3.2639\)  
61710.da8 61710cz1 \([1, 0, 0, 5248070, 1958492900]\) \(9023321954633914439/6156756739584000\) \(-10907070126334170624000\) \([4]\) \(6635520\) \(2.9173\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 61710.da have rank \(0\).

Complex multiplication

The elliptic curves in class 61710.da do not have complex multiplication.

Modular form 61710.2.a.da

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 2 q^{13} + 4 q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.