# Properties

 Label 5586.t Number of curves $4$ Conductor $5586$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5586.t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5586.t1 5586w4 $$[1, 1, 1, -59634, 5580315]$$ $$199350693197713/547428$$ $$64404356772$$ $$$$ $$12288$$ $$1.3072$$
5586.t2 5586w3 $$[1, 1, 1, -10634, -316933]$$ $$1130389181713/295568028$$ $$34773282926172$$ $$$$ $$12288$$ $$1.3072$$
5586.t3 5586w2 $$[1, 1, 1, -3774, 83691]$$ $$50529889873/2547216$$ $$299677415184$$ $$[2, 2]$$ $$6144$$ $$0.96064$$
5586.t4 5586w1 $$[1, 1, 1, 146, 5291]$$ $$2924207/102144$$ $$-12017139456$$ $$$$ $$3072$$ $$0.61407$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5586.t have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5586.t do not have complex multiplication.

## Modular form5586.2.a.t

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 