Properties

Label 28611.r
Number of curves $2$
Conductor $28611$
CM no
Rank $1$
Graph

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

Elliptic curves in class 28611.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28611.r1 28611r2 \([0, 0, 1, -258366, 115628683]\) \(-108394872832/265513259\) \(-4672047720345453459\) \([]\) \(414720\) \(2.2704\)  
28611.r2 28611r1 \([0, 0, 1, 27744, -3536132]\) \(134217728/384659\) \(-6768570469244859\) \([]\) \(138240\) \(1.7211\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 28611.r have rank \(1\).

Complex multiplication

The elliptic curves in class 28611.r do not have complex multiplication.

Modular form 28611.2.a.r

sage: E.q_eigenform(10)
 
\(q - 2 q^{4} + 3 q^{5} - 2 q^{7} + q^{11} + 2 q^{13} + 4 q^{16} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.