Properties

Label 38976bb
Number of curves $6$
Conductor $38976$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 38976bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38976.t5 38976bb1 \([0, -1, 0, -50177, -4615263]\) \(-53297461115137/4513839183\) \(-1183275858788352\) \([2]\) \(196608\) \(1.6372\) \(\Gamma_0(N)\)-optimal
38976.t4 38976bb2 \([0, -1, 0, -818497, -284744735]\) \(231331938231569617/1472026689\) \(385882964361216\) \([2, 2]\) \(393216\) \(1.9838\)  
38976.t3 38976bb3 \([0, -1, 0, -834177, -273251295]\) \(244883173420511137/18418027974129\) \(4828175525250072576\) \([2, 2]\) \(786432\) \(2.3304\)  
38976.t1 38976bb4 \([0, -1, 0, -13095937, -18236817503]\) \(947531277805646290177/38367\) \(10057678848\) \([2]\) \(786432\) \(2.3304\)  
38976.t6 38976bb5 \([0, -1, 0, 798783, -1214162847]\) \(215015459663151503/2552757445339983\) \(-669190047751204503552\) \([2]\) \(1572864\) \(2.6770\)  
38976.t2 38976bb6 \([0, -1, 0, -2718017, 1402989537]\) \(8471112631466271697/1662662681263647\) \(435857045917177479168\) \([2]\) \(1572864\) \(2.6770\)  

Rank

sage: E.rank()
 

The elliptic curves in class 38976bb have rank \(0\).

Complex multiplication

The elliptic curves in class 38976bb do not have complex multiplication.

Modular form 38976.2.a.bb

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.