Properties

Label 31824bk
Number of curves $6$
Conductor $31824$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bk1")
 
E.isogeny_class()
 

Elliptic curves in class 31824bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31824.bg4 31824bk1 \([0, 0, 0, -77619, 8323378]\) \(17319700013617/25857\) \(77208588288\) \([2]\) \(65536\) \(1.3579\) \(\Gamma_0(N)\)-optimal
31824.bg3 31824bk2 \([0, 0, 0, -78339, 8161090]\) \(17806161424897/668584449\) \(1996382467362816\) \([2, 2]\) \(131072\) \(1.7045\)  
31824.bg5 31824bk3 \([0, 0, 0, 31821, 29289778]\) \(1193377118543/124806800313\) \(-372671108825812992\) \([2]\) \(262144\) \(2.0510\)  
31824.bg2 31824bk4 \([0, 0, 0, -200019, -23354030]\) \(296380748763217/92608836489\) \(276528504014770176\) \([2, 2]\) \(262144\) \(2.0510\)  
31824.bg6 31824bk5 \([0, 0, 0, 558141, -158154878]\) \(6439735268725823/7345472585373\) \(-21933463612362412032\) \([2]\) \(524288\) \(2.3976\)  
31824.bg1 31824bk6 \([0, 0, 0, -2905059, -1905520862]\) \(908031902324522977/161726530797\) \(482912833335349248\) \([2]\) \(524288\) \(2.3976\)  

Rank

sage: E.rank()
 

The elliptic curves in class 31824bk have rank \(0\).

Complex multiplication

The elliptic curves in class 31824bk do not have complex multiplication.

Modular form 31824.2.a.bk

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.