Properties

Label 960.p
Number of curves $8$
Conductor $960$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 960.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
960.p1 960o8 \([0, 1, 0, -341345, 76646943]\) \(16778985534208729/81000\) \(21233664000\) \([4]\) \(4608\) \(1.6039\)  
960.p2 960o7 \([0, 1, 0, -29025, 249375]\) \(10316097499609/5859375000\) \(1536000000000000\) \([2]\) \(4608\) \(1.6039\)  
960.p3 960o6 \([0, 1, 0, -21345, 1190943]\) \(4102915888729/9000000\) \(2359296000000\) \([2, 2]\) \(2304\) \(1.2573\)  
960.p4 960o4 \([0, 1, 0, -18465, -971937]\) \(2656166199049/33750\) \(8847360000\) \([2]\) \(1536\) \(1.0546\)  
960.p5 960o5 \([0, 1, 0, -4385, 94815]\) \(35578826569/5314410\) \(1393140695040\) \([4]\) \(1536\) \(1.0546\)  
960.p6 960o2 \([0, 1, 0, -1185, -14625]\) \(702595369/72900\) \(19110297600\) \([2, 2]\) \(768\) \(0.70801\)  
960.p7 960o3 \([0, 1, 0, -865, 31775]\) \(-273359449/1536000\) \(-402653184000\) \([2]\) \(1152\) \(0.91074\)  
960.p8 960o1 \([0, 1, 0, 95, -1057]\) \(357911/2160\) \(-566231040\) \([2]\) \(384\) \(0.36143\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 960.p have rank \(0\).

Complex multiplication

The elliptic curves in class 960.p do not have complex multiplication.

Modular form 960.2.a.p

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.