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Properties

Label 111090.e
Number of curves $2$
Conductor $111090$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 111090.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
111090.e1 111090e1 \([1, 1, 0, -63155998, -193201454348]\) \(188191720927962271801/9422571110400\) \(1394878690993781145600\) \([2]\) \(14598144\) \(3.1282\) \(\Gamma_0(N)\)-optimal
111090.e2 111090e2 \([1, 1, 0, -59770398, -214830698508]\) \(-159520003524722950201/42335913815758080\) \(-6267234638343129581733120\) \([2]\) \(29196288\) \(3.4747\)  

Rank

sage: E.rank()
 

The elliptic curves in class 111090.e have rank \(1\).

Complex multiplication

The elliptic curves in class 111090.e do not have complex multiplication.

Modular form 111090.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} + 4 q^{13} - q^{14} + q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.