# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.c1 450840c2 $$[0, -1, 0, -456716, 114756516]$$ $$1705021456336/68471325$$ $$423099220917484800$$ $$$$ $$7077888$$ $$2.1485$$ $$\Gamma_0(N)$$-optimal*
450840.c2 450840c1 $$[0, -1, 0, 12909, 6554916]$$ $$615962624/48481875$$ $$-18723753648990000$$ $$$$ $$3538944$$ $$1.8019$$ $$\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.c1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 450840.2.a.c

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. 