# Properties

 Label 5577.g Number of curves $6$ Conductor $5577$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5577.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5577.g1 5577g4 [1, 0, 1, -1160020, 480793679]  43008
5577.g2 5577g5 [1, 0, 1, -508525, -135200167]  86016
5577.g3 5577g3 [1, 0, 1, -80110, 5834051] [2, 2] 43008
5577.g4 5577g2 [1, 0, 1, -72505, 7507151] [2, 2] 21504
5577.g5 5577g1 [1, 0, 1, -4060, 142469]  10752 $$\Gamma_0(N)$$-optimal
5577.g6 5577g6 [1, 0, 1, 226625, 39820289]  86016

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 5577.g do not have complex multiplication.

## Modular form5577.2.a.g

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} - 3q^{8} + q^{9} + 2q^{10} + q^{11} - q^{12} + 2q^{15} - q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 