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Properties

Label 86394w
Number of curves $2$
Conductor $86394$
CM no
Rank $1$
Graph

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

Elliptic curves in class 86394w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86394.w2 86394w1 \([1, 0, 1, 16855660, 1534247570]\) \(298954383299125345007/173578967063986176\) \(-307505728470842413940736\) \([]\) \(12441600\) \(3.1964\) \(\Gamma_0(N)\)-optimal
86394.w1 86394w2 \([1, 0, 1, -237883220, 1499323032722]\) \(-840347716046483516416273/61875690788436630336\) \(-109616560648853585274674496\) \([]\) \(37324800\) \(3.7457\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86394w have rank \(1\).

Complex multiplication

The elliptic curves in class 86394w do not have complex multiplication.

Modular form 86394.2.a.w

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