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Properties

Label 118698.f
Number of curves $3$
Conductor $118698$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 118698.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
118698.f1 118698i3 \([1, 0, 0, -130692214242, -18185389699141044]\) \(246872582520723794163685866669320846113/220813028718485844638509857144\) \(220813028718485844638509857144\) \([]\) \(621038016\) \(4.9296\)  
118698.f2 118698i2 \([1, 0, 0, -1989829962, -12444227211996]\) \(871307816553819317720750687651233/437331845679894053147718065664\) \(437331845679894053147718065664\) \([3]\) \(207012672\) \(4.3803\)  
118698.f3 118698i1 \([1, 0, 0, -1080971082, 13678914712356]\) \(139690200244171980376257072833953/5481890282879036260614144\) \(5481890282879036260614144\) \([9]\) \(69004224\) \(3.8310\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 118698.f have rank \(1\).

Complex multiplication

The elliptic curves in class 118698.f do not have complex multiplication.

Modular form 118698.2.a.f

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.