# Properties

 Label 405600.bi Number of curves $2$ Conductor $405600$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bi1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.bi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.bi1 405600bi2 $$[0, -1, 0, -5578408, -5067697688]$$ $$497169541448/190125$$ $$7341576489000000000$$ $$$$ $$15482880$$ $$2.5859$$ $$\Gamma_0(N)$$-optimal*
405600.bi2 405600bi1 $$[0, -1, 0, -297158, -103322688]$$ $$-601211584/609375$$ $$-2941336734375000000$$ $$$$ $$7741440$$ $$2.2393$$ $$\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 405600.bi1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 405600.2.a.bi

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{7} + q^{9} + 4q^{11} - 6q^{19} + O(q^{20})$$ ## 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. 