Properties

Label 630.a
Number of curves $8$
Conductor $630$
CM no
Rank $0$
Graph

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E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 630.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
630.a1 630c7 \([1, -1, 0, -17287200, 27669604050]\) \(783736670177727068275201/360150\) \(262549350\) \([2]\) \(16384\) \(2.3437\)  
630.a2 630c5 \([1, -1, 0, -1080450, 432540000]\) \(191342053882402567201/129708022500\) \(94557148402500\) \([2, 2]\) \(8192\) \(1.9972\)  
630.a3 630c8 \([1, -1, 0, -1073700, 438205950]\) \(-187778242790732059201/4984939585440150\) \(-3634020957785869350\) \([2]\) \(16384\) \(2.3437\)  
630.a4 630c3 \([1, -1, 0, -135630, -19186524]\) \(378499465220294881/120530818800\) \(87866966905200\) \([2]\) \(4096\) \(1.6506\)  
630.a5 630c4 \([1, -1, 0, -67950, 6682500]\) \(47595748626367201/1215506250000\) \(886104056250000\) \([2, 2]\) \(4096\) \(1.6506\)  
630.a6 630c2 \([1, -1, 0, -9630, -210924]\) \(135487869158881/51438240000\) \(37498476960000\) \([2, 2]\) \(2048\) \(1.3040\)  
630.a7 630c1 \([1, -1, 0, 1890, -24300]\) \(1023887723039/928972800\) \(-677221171200\) \([2]\) \(1024\) \(0.95744\) \(\Gamma_0(N)\)-optimal
630.a8 630c6 \([1, -1, 0, 11430, 21304296]\) \(226523624554079/269165039062500\) \(-196221313476562500\) \([2]\) \(8192\) \(1.9972\)  

Rank

sage: E.rank()
 

The elliptic curves in class 630.a have rank \(0\).

Complex multiplication

The elliptic curves in class 630.a do not have complex multiplication.

Modular form 630.2.a.a

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.