Properties

Label 47040.hb
Number of curves $6$
Conductor $47040$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hb1")
 
E.isogeny_class()
 

Elliptic curves in class 47040.hb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.hb1 47040hc6 \([0, 1, 0, -52684865, -147207093825]\) \(524388516989299201/3150\) \(97149085286400\) \([2]\) \(2359296\) \(2.7494\)  
47040.hb2 47040hc4 \([0, 1, 0, -3292865, -2300844225]\) \(128031684631201/9922500\) \(306019618652160000\) \([2, 2]\) \(1179648\) \(2.4029\)  
47040.hb3 47040hc5 \([0, 1, 0, -3073345, -2620597057]\) \(-104094944089921/35880468750\) \(-1106588799590400000000\) \([4]\) \(2359296\) \(2.7494\)  
47040.hb4 47040hc3 \([0, 1, 0, -1160385, 454345023]\) \(5602762882081/345888060\) \(10667526552535695360\) \([2]\) \(1179648\) \(2.4029\)  
47040.hb5 47040hc2 \([0, 1, 0, -219585, -30919617]\) \(37966934881/8643600\) \(266577090025881600\) \([2, 2]\) \(589824\) \(2.0563\)  
47040.hb6 47040hc1 \([0, 1, 0, 31295, -2971585]\) \(109902239/188160\) \(-5803038694440960\) \([2]\) \(294912\) \(1.7097\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 47040.hb have rank \(1\).

Complex multiplication

The elliptic curves in class 47040.hb do not have complex multiplication.

Modular form 47040.2.a.hb

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.