Properties

Label 53235bc
Number of curves $4$
Conductor $53235$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 53235bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53235.bg3 53235bc1 \([1, -1, 0, -3834, -85617]\) \(1771561/105\) \(369468094905\) \([2]\) \(61440\) \(0.97267\) \(\Gamma_0(N)\)-optimal
53235.bg2 53235bc2 \([1, -1, 0, -11439, 366120]\) \(47045881/11025\) \(38794149965025\) \([2, 2]\) \(122880\) \(1.3192\)  
53235.bg4 53235bc3 \([1, -1, 0, 26586, 2259765]\) \(590589719/972405\) \(-3421644026915205\) \([2]\) \(245760\) \(1.6658\)  
53235.bg1 53235bc4 \([1, -1, 0, -171144, 27292383]\) \(157551496201/13125\) \(46183511863125\) \([2]\) \(245760\) \(1.6658\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53235bc have rank \(1\).

Complex multiplication

The elliptic curves in class 53235bc do not have complex multiplication.

Modular form 53235.2.a.bc

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - q^{7} - 3 q^{8} + q^{10} - q^{14} - q^{16} - 2 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.