Properties

Label 333270.dy
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dy1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.dy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.dy1 333270dy3 \([1, -1, 1, -8777003, 9346920581]\) \(692895692874169/51420783750\) \(5549236526485334733750\) \([2]\) \(25952256\) \(2.9181\)  
333270.dy2 333270dy2 \([1, -1, 1, -1778333, -739562623]\) \(5763259856089/1143116100\) \(123362989700316704100\) \([2, 2]\) \(12976128\) \(2.5715\)  
333270.dy3 333270dy1 \([1, -1, 1, -1683113, -840000679]\) \(4886171981209/270480\) \(29189704750148880\) \([2]\) \(6488064\) \(2.2249\) \(\Gamma_0(N)\)-optimal
333270.dy4 333270dy4 \([1, -1, 1, 3696817, -4399152883]\) \(51774168853511/107398242630\) \(-11590221062757146343030\) \([2]\) \(25952256\) \(2.9181\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.dy have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.dy do not have complex multiplication.

Modular form 333270.2.a.dy

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + 4 q^{11} + 2 q^{13} + q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.