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Properties

Label 56355b
Number of curves $8$
Conductor $56355$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 56355b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
56355.b6 56355b1 \([1, 1, 1, -31796, 2168948]\) \(147281603041/5265\) \(127084300785\) \([2]\) \(122880\) \(1.2197\) \(\Gamma_0(N)\)-optimal
56355.b5 56355b2 \([1, 1, 1, -33241, 1959134]\) \(168288035761/27720225\) \(669098843633025\) \([2, 2]\) \(245760\) \(1.5663\)  
56355.b7 56355b3 \([1, 1, 1, 60684, 11126214]\) \(1023887723039/2798036865\) \(-67537807893481185\) \([2]\) \(491520\) \(1.9129\)  
56355.b4 56355b4 \([1, 1, 1, -150286, -20607142]\) \(15551989015681/1445900625\) \(34900526103080625\) \([2, 2]\) \(491520\) \(1.9129\)  
56355.b8 56355b5 \([1, 1, 1, 174839, -97206592]\) \(24487529386319/183539412225\) \(-4430195226800381025\) \([2]\) \(983040\) \(2.2594\)  
56355.b2 56355b6 \([1, 1, 1, -2348131, -1385908456]\) \(59319456301170001/594140625\) \(14341110331640625\) \([2, 2]\) \(983040\) \(2.2594\)  
56355.b3 56355b7 \([1, 1, 1, -2291776, -1455518152]\) \(-55150149867714721/5950927734375\) \(-143640928802490234375\) \([2]\) \(1966080\) \(2.6060\)  
56355.b1 56355b8 \([1, 1, 1, -37570006, -88651625956]\) \(242970740812818720001/24375\) \(588353244375\) \([2]\) \(1966080\) \(2.6060\)  

Rank

sage: E.rank()
 

The elliptic curves in class 56355b have rank \(2\).

Complex multiplication

The elliptic curves in class 56355b do not have complex multiplication.

Modular form 56355.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + 3 q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} + q^{13} + q^{15} - q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.