# Properties

 Label 405600b Number of curves $4$ Conductor $405600$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 405600b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.b3 405600b1 $$[0, -1, 0, -9858, 206712]$$ $$21952/9$$ $$43441281000000$$ $$[2, 2]$$ $$1179648$$ $$1.3144$$ $$\Gamma_0(N)$$-optimal*
405600.b1 405600b2 $$[0, -1, 0, -136608, 19472712]$$ $$7301384/3$$ $$115843416000000$$ $$[2]$$ $$2359296$$ $$1.6610$$ $$\Gamma_0(N)$$-optimal*
405600.b4 405600b3 $$[0, -1, 0, 32392, 1474212]$$ $$97336/81$$ $$-3127772232000000$$ $$[2]$$ $$2359296$$ $$1.6610$$
405600.b2 405600b4 $$[0, -1, 0, -73233, -7461663]$$ $$140608/3$$ $$926747328000000$$ $$[2]$$ $$2359296$$ $$1.6610$$
*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 405600b1.

## Rank

sage: E.rank()

The elliptic curves in class 405600b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405600b do not have complex multiplication.

## Modular form 405600.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 4q^{7} + q^{9} - 4q^{11} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.