Properties

Label 240.b
Number of curves $8$
Conductor $240$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 240.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
240.b1 240b8 \([0, -1, 0, -85336, 9623536]\) \(16778985534208729/81000\) \(331776000\) \([2]\) \(576\) \(1.2573\)  
240.b2 240b7 \([0, -1, 0, -7256, 34800]\) \(10316097499609/5859375000\) \(24000000000000\) \([2]\) \(576\) \(1.2573\)  
240.b3 240b6 \([0, -1, 0, -5336, 151536]\) \(4102915888729/9000000\) \(36864000000\) \([2, 2]\) \(288\) \(0.91074\)  
240.b4 240b4 \([0, -1, 0, -4616, -119184]\) \(2656166199049/33750\) \(138240000\) \([2]\) \(192\) \(0.70801\)  
240.b5 240b5 \([0, -1, 0, -1096, 12400]\) \(35578826569/5314410\) \(21767823360\) \([2]\) \(192\) \(0.70801\)  
240.b6 240b2 \([0, -1, 0, -296, -1680]\) \(702595369/72900\) \(298598400\) \([2, 2]\) \(96\) \(0.36143\)  
240.b7 240b3 \([0, -1, 0, -216, 4080]\) \(-273359449/1536000\) \(-6291456000\) \([2]\) \(144\) \(0.56417\)  
240.b8 240b1 \([0, -1, 0, 24, -144]\) \(357911/2160\) \(-8847360\) \([2]\) \(48\) \(0.014860\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 240.b have rank \(0\).

Complex multiplication

The elliptic curves in class 240.b do not have complex multiplication.

Modular form 240.2.a.b

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.