Properties

Label 16320bt
Number of curves $8$
Conductor $16320$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bt1")
 
E.isogeny_class()
 

Elliptic curves in class 16320bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16320.m6 16320bt1 \([0, -1, 0, -5121, -171615]\) \(-56667352321/16711680\) \(-4380866641920\) \([2]\) \(24576\) \(1.1402\) \(\Gamma_0(N)\)-optimal
16320.m5 16320bt2 \([0, -1, 0, -87041, -9854559]\) \(278202094583041/16646400\) \(4363753881600\) \([2, 2]\) \(49152\) \(1.4867\)  
16320.m2 16320bt3 \([0, -1, 0, -1392641, -632103519]\) \(1139466686381936641/4080\) \(1069547520\) \([2]\) \(98304\) \(1.8333\)  
16320.m4 16320bt4 \([0, -1, 0, -92161, -8624735]\) \(330240275458561/67652010000\) \(17734568509440000\) \([2, 2]\) \(98304\) \(1.8333\)  
16320.m3 16320bt5 \([0, -1, 0, -462081, 113374881]\) \(41623544884956481/2962701562500\) \(776654438400000000\) \([2, 2]\) \(196608\) \(2.1799\)  
16320.m7 16320bt6 \([0, -1, 0, 195839, -51997535]\) \(3168685387909439/6278181696900\) \(-1645787662752153600\) \([2]\) \(196608\) \(2.1799\)  
16320.m1 16320bt7 \([0, -1, 0, -7262081, 7534894881]\) \(161572377633716256481/914742821250\) \(239794342133760000\) \([2]\) \(393216\) \(2.5265\)  
16320.m8 16320bt8 \([0, -1, 0, 419199, 493911585]\) \(31077313442863199/420227050781250\) \(-110160000000000000000\) \([2]\) \(393216\) \(2.5265\)  

Rank

sage: E.rank()
 

The elliptic curves in class 16320bt have rank \(1\).

Complex multiplication

The elliptic curves in class 16320bt do not have complex multiplication.

Modular form 16320.2.a.bt

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.