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Properties

Label 272734.bm
Number of curves $2$
Conductor $272734$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bm1")
 
E.isogeny_class()
 

Elliptic curves in class 272734.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
272734.bm1 272734bm2 \([1, 1, 0, -2804540, 1797762212]\) \(11704814052625/66001628\) \(13756216397508784892\) \([2]\) \(8847360\) \(2.5144\)  
272734.bm2 272734bm1 \([1, 1, 0, -77200, 59355696]\) \(-244140625/7169008\) \(-1494181710237081712\) \([2]\) \(4423680\) \(2.1678\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 272734.bm have rank \(1\).

Complex multiplication

The elliptic curves in class 272734.bm do not have complex multiplication.

Modular form 272734.2.a.bm

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - q^{8} + q^{9} + 2 q^{12} - 2 q^{13} + q^{16} - 4 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.