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Properties

Label 874e
Number of curves $2$
Conductor $874$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 874e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
874.d2 874e1 \([1, 1, 1, -410, 903]\) \(7623273198241/3913296544\) \(3913296544\) \([5]\) \(400\) \(0.53285\) \(\Gamma_0(N)\)-optimal
874.d1 874e2 \([1, 1, 1, -142320, -20724857]\) \(318802492415278298881/113900554\) \(113900554\) \([]\) \(2000\) \(1.3376\)  

Rank

sage: E.rank()
 

The elliptic curves in class 874e have rank \(1\).

Complex multiplication

The elliptic curves in class 874e do not have complex multiplication.

Modular form 874.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.