Properties

Label 422370fn
Number of curves $2$
Conductor $422370$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("fn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 422370fn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422370.fn2 422370fn1 \([1, -1, 1, 12928, -1573729]\) \(6967871/35100\) \(-1203805298439900\) \([2]\) \(2515968\) \(1.5759\) \(\Gamma_0(N)\)-optimal*
422370.fn1 422370fn2 \([1, -1, 1, -149522, -19898089]\) \(10779215329/1232010\) \(42253565975240490\) \([2]\) \(5031936\) \(1.9225\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 422370fn1.

Rank

sage: E.rank()
 

The elliptic curves in class 422370fn have rank \(0\).

Complex multiplication

The elliptic curves in class 422370fn do not have complex multiplication.

Modular form 422370.2.a.fn

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + q^{10} - 4q^{11} + q^{13} + 2q^{14} + q^{16} - 8q^{17} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.