# Properties

 Label 111573p Number of curves 2 Conductor 111573 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("111573.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 111573p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111573.bd2 111573p1 [1, -1, 0, -10152, -5722893]  460800 $$\Gamma_0(N)$$-optimal
111573.bd1 111573p2 [1, -1, 0, -545967, -153929322]  921600

## Rank

sage: E.rank()

The elliptic curves in class 111573p have rank $$1$$.

## Modular form 111573.2.a.bd

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3q^{8} - q^{11} - 2q^{13} - q^{16} - 2q^{19} + O(q^{20})$$

## 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. 