Properties

Label 215985.bu
Number of curves $4$
Conductor $215985$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bu1")
 
E.isogeny_class()
 

Elliptic curves in class 215985.bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
215985.bu1 215985bp4 \([1, 0, 1, -204009, -35482163]\) \(530044731605089/26309115\) \(46608202078515\) \([2]\) \(1474560\) \(1.6941\)  
215985.bu2 215985bp3 \([1, 0, 1, -64859, 5904677]\) \(17032120495489/1339001685\) \(2372123164080285\) \([2]\) \(1474560\) \(1.6941\)  
215985.bu3 215985bp2 \([1, 0, 1, -13434, -492593]\) \(151334226289/28676025\) \(50801327525025\) \([2, 2]\) \(737280\) \(1.3475\)  
215985.bu4 215985bp1 \([1, 0, 1, 1691, -44893]\) \(302111711/669375\) \(-1185838644375\) \([2]\) \(368640\) \(1.0009\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 215985.bu have rank \(0\).

Complex multiplication

The elliptic curves in class 215985.bu do not have complex multiplication.

Modular form 215985.2.a.bu

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{10} - q^{12} + 6 q^{13} - q^{14} - q^{15} - q^{16} + q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.