# Properties

 Label 450840bl Number of curves $4$ Conductor $450840$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bl1")

sage: E.isogeny_class()

## Elliptic curves in class 450840bl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.bl4 450840bl1 $$[0, 1, 0, -5876, 613440]$$ $$-3631696/24375$$ $$-150618430560000$$ $$[2]$$ $$1769472$$ $$1.4038$$ $$\Gamma_0(N)$$-optimal*
450840.bl3 450840bl2 $$[0, 1, 0, -150376, 22346240]$$ $$15214885924/38025$$ $$939859006694400$$ $$[2, 2]$$ $$3538944$$ $$1.7504$$ $$\Gamma_0(N)$$-optimal*
450840.bl1 450840bl3 $$[0, 1, 0, -2404576, 1434377120]$$ $$31103978031362/195$$ $$9639579555840$$ $$[2]$$ $$7077888$$ $$2.0970$$ $$\Gamma_0(N)$$-optimal*
450840.bl2 450840bl4 $$[0, 1, 0, -208176, 3526560]$$ $$20183398562/11567205$$ $$571810219672872960$$ $$[2]$$ $$7077888$$ $$2.0970$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 450840bl1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 450840.2.a.bl

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - 4q^{7} + q^{9} - 4q^{11} + q^{13} - q^{15} + O(q^{20})$$

## 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 Cremona 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 Cremona labels.