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Properties

Label 272734r
Number of curves $2$
Conductor $272734$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 272734r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
272734.r2 272734r1 \([1, -1, 0, -212782334, -1194666915436]\) \(-1753396709868750829527/68729140412416\) \(-41762947598328914771968\) \([2]\) \(71884800\) \(3.4246\) \(\Gamma_0(N)\)-optimal
272734.r1 272734r2 \([1, -1, 0, -3404549374, -76459725485676]\) \(7182115611944007873394647/11034394624\) \(6704999388849405952\) \([2]\) \(143769600\) \(3.7712\)  

Rank

sage: E.rank()
 

The elliptic curves in class 272734r have rank \(0\).

Complex multiplication

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

Modular form 272734.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4 q^{5} - q^{8} - 3 q^{9} + 4 q^{10} + 2 q^{13} + q^{16} + 6 q^{17} + 3 q^{18} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.