# Properties

 Label 111090o Number of curves $4$ Conductor $111090$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 111090o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111090.p3 111090o1 [1, 1, 0, -1862, 18756] [2] 180224 $$\Gamma_0(N)$$-optimal
111090.p2 111090o2 [1, 1, 0, -12442, -525056] [2, 2] 360448
111090.p4 111090o3 [1, 1, 0, 3428, -1753394] [2] 720896
111090.p1 111090o4 [1, 1, 0, -197592, -33889086] [2] 720896

## Rank

sage: E.rank()

The elliptic curves in class 111090o have rank $$0$$.

## Complex multiplication

The elliptic curves in class 111090o do not have complex multiplication.

## Modular form 111090.2.a.o

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - 2q^{13} - q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} + O(q^{20})$$

## 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.