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Properties

Label 69454e
Number of curves $2$
Conductor $69454$
CM no
Rank $1$
Graph

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

Elliptic curves in class 69454e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69454.a2 69454e1 \([1, -1, 0, -2341675, -1269022331]\) \(801581275315909089/70810888830976\) \(125445809028292673536\) \([]\) \(4939200\) \(2.5967\) \(\Gamma_0(N)\)-optimal
69454.a1 69454e2 \([1, -1, 0, -1162968835, 15265410956989]\) \(98191033604529537629349729/10906239337336\) \(19321068266690301496\) \([]\) \(34574400\) \(3.5697\)  

Rank

sage: E.rank()
 

The elliptic curves in class 69454e have rank \(1\).

Complex multiplication

The elliptic curves in class 69454e do not have complex multiplication.

Modular form 69454.2.a.e

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.