Properties

Label 216384.bd
Number of curves $6$
Conductor $216384$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 216384.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216384.bd1 216384hm5 \([0, -1, 0, -248164289, 1504792275393]\) \(54804145548726848737/637608031452\) \(19664456198991733850112\) \([2]\) \(37748736\) \(3.4286\)  
216384.bd2 216384hm4 \([0, -1, 0, -55551169, -159340362047]\) \(614716917569296417/19093020912\) \(588847465699970383872\) \([2]\) \(18874368\) \(3.0821\)  
216384.bd3 216384hm3 \([0, -1, 0, -15912129, 22233837249]\) \(14447092394873377/1439452851984\) \(44394135835919152840704\) \([2, 2]\) \(18874368\) \(3.0821\)  
216384.bd4 216384hm2 \([0, -1, 0, -3619009, -2266350911]\) \(169967019783457/26337394944\) \(812271056392462270464\) \([2, 2]\) \(9437184\) \(2.7355\)  
216384.bd5 216384hm1 \([0, -1, 0, 395071, -195888447]\) \(221115865823/664731648\) \(-20500975099721023488\) \([2]\) \(4718592\) \(2.3889\) \(\Gamma_0(N)\)-optimal
216384.bd6 216384hm6 \([0, -1, 0, 19650111, 107490751425]\) \(27207619911317663/177609314617308\) \(-5477645223306636530024448\) \([2]\) \(37748736\) \(3.4286\)  

Rank

sage: E.rank()
 

The elliptic curves in class 216384.bd have rank \(0\).

Complex multiplication

The elliptic curves in class 216384.bd do not have complex multiplication.

Modular form 216384.2.a.bd

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.