Properties

Label 11a
Number of curves $3$
Conductor $11$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 11a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
11.a2 11a1 \([0, -1, 1, -10, -20]\) \(-122023936/161051\) \(-161051\) \([5]\) \(1\) \(-0.30801\) \(\Gamma_0(N)\)-optimal
11.a1 11a2 \([0, -1, 1, -7820, -263580]\) \(-52893159101157376/11\) \(-11\) \([]\) \(5\) \(0.49671\)  
11.a3 11a3 \([0, -1, 1, 0, 0]\) \(-4096/11\) \(-11\) \([5]\) \(5\) \(-1.1127\)  

Rank

sage: E.rank()
 

The elliptic curves in class 11a have rank \(0\).

Complex multiplication

The elliptic curves in class 11a do not have complex multiplication.

Modular form 11.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{9} - 2q^{10} + q^{11} - 2q^{12} + 4q^{13} + 4q^{14} - q^{15} - 4q^{16} - 2q^{17} + 4q^{18} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 5 & 5 \\ 5 & 1 & 25 \\ 5 & 25 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.