# Properties

 Label 414736.g Number of curves $2$ Conductor $414736$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 414736.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.g1 414736g2 $$[0, 1, 0, -14109664, -19791928668]$$ $$1431644/49$$ $$10632485543889473395712$$ $$$$ $$23740416$$ $$2.9984$$
414736.g2 414736g1 $$[0, 1, 0, -2186004, 816925276]$$ $$21296/7$$ $$379731626567481192704$$ $$$$ $$11870208$$ $$2.6518$$ $$\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 0 curves highlighted, and conditionally curve 414736.g1.

## Rank

sage: E.rank()

The elliptic curves in class 414736.g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 414736.g do not have complex multiplication.

## Modular form 414736.2.a.g

sage: E.q_eigenform(10)

$$q - 2 q^{3} - 2 q^{5} + q^{9} + 2 q^{13} + 4 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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. 