# Properties

 Label 55545.t Number of curves $4$ Conductor $55545$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("t1")

sage: E.isogeny_class()

## Elliptic curves in class 55545.t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55545.t1 55545g4 [1, 0, 1, -59524, 5584241] [2] 180224
55545.t2 55545g2 [1, 0, 1, -3979, 74177] [2, 2] 90112
55545.t3 55545g1 [1, 0, 1, -1334, -17869] [2] 45056 $$\Gamma_0(N)$$-optimal
55545.t4 55545g3 [1, 0, 1, 9246, 465637] [2] 180224

## Rank

sage: E.rank()

The elliptic curves in class 55545.t have rank $$1$$.

## Complex multiplication

The elliptic curves in class 55545.t do not have complex multiplication.

## Modular form 55545.2.a.t

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{7} - 3q^{8} + q^{9} - q^{10} - q^{12} - 6q^{13} - q^{14} - q^{15} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.