# Properties

 Label 111090.bn Number of curves $6$ Conductor $111090$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("111090.bn1")

sage: E.isogeny_class()

## Elliptic curves in class 111090.bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111090.bn1 111090br6 [1, 1, 1, -8887211, 10193858939] [2] 2883584
111090.bn2 111090br4 [1, 1, 1, -555461, 159099239] [2, 2] 1441792
111090.bn3 111090br5 [1, 1, 1, -518431, 181272803] [2] 2883584
111090.bn4 111090br3 [1, 1, 1, -195741, -31586217] [2] 1441792
111090.bn5 111090br2 [1, 1, 1, -37041, 2121663] [2, 2] 720896
111090.bn6 111090br1 [1, 1, 1, 5279, 208799] [2] 360448 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 111090.bn have rank $$1$$.

## Modular form 111090.2.a.bn

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 2q^{13} - q^{14} + q^{15} + q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.