# Properties

 Label 450840.k Number of curves $2$ Conductor $450840$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.k1 450840k2 $$[0, -1, 0, -2394410795036, -1425618290334931260]$$ $$245689277968779868090419995701456/93342399137270122585475925$$ $$576783001549157902219618185606739200$$ $$$$ $$8918138880$$ $$5.8281$$ $$\Gamma_0(N)$$-optimal*
450840.k2 450840k1 $$[0, -1, 0, -127620618411, -29061556541257860]$$ $$-595213448747095198927846967296/600281130562949295663181875$$ $$-231829235333779159481143252277790000$$ $$$$ $$4459069440$$ $$5.4816$$ $$\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 450840.k1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 450840.k do not have complex multiplication.

## Modular form 450840.2.a.k

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + 2q^{7} + q^{9} + q^{13} + q^{15} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 