# Properties

 Label 57222bh Number of curves 4 Conductor 57222 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("57222.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 57222bh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57222.bm3 57222bh1 [1, -1, 1, -14360, -621129]  147456 $$\Gamma_0(N)$$-optimal
57222.bm4 57222bh2 [1, -1, 1, 11650, -2639505]  294912
57222.bm1 57222bh3 [1, -1, 1, -209435, 36802059]  442368
57222.bm2 57222bh4 [1, -1, 1, -105395, 73299291]  884736

## Rank

sage: E.rank()

The elliptic curves in class 57222bh have rank $$1$$.

## Modular form 57222.2.a.bm

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 2q^{7} + q^{8} - q^{11} - 4q^{13} - 2q^{14} + q^{16} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 