Properties

Label 31680.bk
Number of curves $4$
Conductor $31680$
CM no
Rank $1$
Graph

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

Elliptic curves in class 31680.bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31680.bk1 31680x4 \([0, 0, 0, -8999868, 10392076208]\) \(6749703004355978704/5671875\) \(67744512000000\) \([2]\) \(442368\) \(2.3894\)  
31680.bk2 31680x3 \([0, 0, 0, -562368, 162451208]\) \(-26348629355659264/24169921875\) \(-18042750000000000\) \([2]\) \(221184\) \(2.0429\)  
31680.bk3 31680x2 \([0, 0, 0, -113628, 13575152]\) \(13584145739344/1195803675\) \(14282602562764800\) \([2]\) \(147456\) \(1.8401\)  
31680.bk4 31680x1 \([0, 0, 0, 7872, 987752]\) \(72268906496/606436875\) \(-452702701440000\) \([2]\) \(73728\) \(1.4936\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 31680.bk have rank \(1\).

Complex multiplication

The elliptic curves in class 31680.bk do not have complex multiplication.

Modular form 31680.2.a.bk

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.