# Properties

 Label 405600q Number of curves $2$ Conductor $405600$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 405600q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.q2 405600q1 $$[0, -1, 0, 311242, -110116488]$$ $$314432/675$$ $$-7158037076775000000$$ $$[2]$$ $$10063872$$ $$2.3022$$ $$\Gamma_0(N)$$-optimal*
405600.q1 405600q2 $$[0, -1, 0, -2435008, -1192138988]$$ $$18821096/3645$$ $$309227201716680000000$$ $$[2]$$ $$20127744$$ $$2.6487$$ $$\Gamma_0(N)$$-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 405600q1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 405600.2.a.q

sage: E.q_eigenform(10)

$$q - q^{3} - 4q^{7} + q^{9} + 4q^{11} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.