Properties

Label 138720.bp
Number of curves $4$
Conductor $138720$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bp1")
 
E.isogeny_class()
 

Elliptic curves in class 138720.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
138720.bp1 138720f2 \([0, 1, 0, -1040785, 408338975]\) \(1261112198464/675\) \(66735550771200\) \([2]\) \(1966080\) \(1.9822\)  
138720.bp2 138720f4 \([0, 1, 0, -143440, -11688712]\) \(26410345352/10546875\) \(130342872600000000\) \([2]\) \(1966080\) \(1.9822\)  
138720.bp3 138720f1 \([0, 1, 0, -65410, 6289400]\) \(20034997696/455625\) \(703851512040000\) \([2, 2]\) \(983040\) \(1.6357\) \(\Gamma_0(N)\)-optimal
138720.bp4 138720f3 \([0, 1, 0, 6840, 19496700]\) \(2863288/13286025\) \(-164194480728691200\) \([2]\) \(1966080\) \(1.9822\)  

Rank

sage: E.rank()
 

The elliptic curves in class 138720.bp have rank \(0\).

Complex multiplication

The elliptic curves in class 138720.bp do not have complex multiplication.

Modular form 138720.2.a.bp

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + 4 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.