# Properties

 Label 450840.p Number of curves $4$ Conductor $450840$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.p1 450840p4 $$[0, -1, 0, -26079456, -38206192644]$$ $$79364416584061444/20404090514925$$ $$504325266110717662540800$$ $$$$ $$56623104$$ $$3.2575$$
450840.p2 450840p2 $$[0, -1, 0, -9172956, 10207260756]$$ $$13813960087661776/714574355625$$ $$4415510480519417760000$$ $$[2, 2]$$ $$28311552$$ $$2.9109$$
450840.p3 450840p1 $$[0, -1, 0, -9055911, 10492288740]$$ $$212670222886967296/616241925$$ $$237993311766085200$$ $$$$ $$14155776$$ $$2.5643$$ $$\Gamma_0(N)$$-optimal*
450840.p4 450840p3 $$[0, -1, 0, 5860824, 40377050460]$$ $$900753985478876/29018422265625$$ $$-717244589780643600000000$$ $$$$ $$56623104$$ $$3.2575$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 450840.p1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.p have rank $$0$$.

## Complex multiplication

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

## Modular form 450840.2.a.p

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 