Properties

Label 216384l
Number of curves $6$
Conductor $216384$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("216384.fi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 216384l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
216384.fi5 216384l1 [0, 1, 0, 395071, 195888447] [2] 4718592 \(\Gamma_0(N)\)-optimal
216384.fi4 216384l2 [0, 1, 0, -3619009, 2266350911] [2, 2] 9437184  
216384.fi2 216384l3 [0, 1, 0, -55551169, 159340362047] [2] 18874368  
216384.fi3 216384l4 [0, 1, 0, -15912129, -22233837249] [2, 2] 18874368  
216384.fi6 216384l5 [0, 1, 0, 19650111, -107490751425] [2] 37748736  
216384.fi1 216384l6 [0, 1, 0, -248164289, -1504792275393] [2] 37748736  

Rank

sage: E.rank()
 

The elliptic curves in class 216384l have rank \(1\).

Modular form 216384.2.a.fi

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.