# Properties

 Label 5577d Number of curves 4 Conductor 5577 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5577.a1")

sage: E.isogeny_class()

## Elliptic curves in class 5577d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5577.a3 5577d1 [1, 1, 1, -1102, -14422]  3456 $$\Gamma_0(N)$$-optimal
5577.a2 5577d2 [1, 1, 1, -1947, 9576] [2, 2] 6912
5577.a1 5577d3 [1, 1, 1, -24762, 1487988]  13824
5577.a4 5577d4 [1, 1, 1, 7348, 83936]  13824

## Rank

sage: E.rank()

The elliptic curves in class 5577d have rank $$1$$.

## Modular form5577.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} - 4q^{7} + 3q^{8} + q^{9} - 2q^{10} - q^{11} + q^{12} + 4q^{14} - 2q^{15} - q^{16} - 2q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 