# Properties

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

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 $$[1, -1, 1, 12928, -1573729]$$ $$6967871/35100$$ $$-1203805298439900$$ $$$$ $$2515968$$ $$1.5759$$ $$\Gamma_0(N)$$-optimal*
422370.fn1 422370fn2 $$[1, -1, 1, -149522, -19898089]$$ $$10779215329/1232010$$ $$42253565975240490$$ $$$$ $$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)

$$q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + q^{10} - 4q^{11} + q^{13} + 2q^{14} + q^{16} - 8q^{17} + O(q^{20})$$ ## 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. 