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Properties

Label 111573.w
Number of curves $2$
Conductor $111573$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 111573.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
111573.w1 111573s2 \([1, -1, 0, -46314, -3370991]\) \(128100283921/16500407\) \(1415175903311247\) \([2]\) \(921600\) \(1.6359\)  
111573.w2 111573s1 \([1, -1, 0, 4401, -277376]\) \(109902239/448063\) \(-38428625473623\) \([2]\) \(460800\) \(1.2893\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 111573.w have rank \(1\).

Complex multiplication

The elliptic curves in class 111573.w do not have complex multiplication.

Modular form 111573.2.a.w

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