# Properties

 Label 28322.r Number of curves 4 Conductor 28322 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("28322.r1")

sage: E.isogeny_class()

## Elliptic curves in class 28322.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28322.r1 28322ba4 [1, 0, 0, -1600488, 538413406]  995328
28322.r2 28322ba3 [1, 0, 0, -1458878, 678012544]  497664
28322.r3 28322ba2 [1, 0, 0, -609218, -183032900]  331776
28322.r4 28322ba1 [1, 0, 0, -42778, -2111964]  165888 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 28322.r have rank $$1$$.

## Modular form 28322.2.a.r

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + q^{8} + q^{9} - 6q^{11} - 2q^{12} - 2q^{13} + q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels. 