# Properties

 Label 86190.bm Number of curves 8 Conductor 86190 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("86190.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 86190.bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86190.bm1 86190bm8 [1, 0, 1, -1113879001018, -452486084448911992] [2] 557383680
86190.bm2 86190bm6 [1, 0, 1, -69617438518, -7070099217161992] [2, 2] 278691840
86190.bm3 86190bm7 [1, 0, 1, -69491198898, -7097017240838744] [4] 557383680
86190.bm4 86190bm5 [1, 0, 1, -13752081958, -620648852140744] [2] 185794560
86190.bm5 86190bm3 [1, 0, 1, -4358980838, -110049775142344] [2] 139345920
86190.bm6 86190bm2 [1, 0, 1, -939177958, -7792278335944] [2, 2] 92897280
86190.bm7 86190bm1 [1, 0, 1, -354248678, 2471124782648] [2] 46448640 $$\Gamma_0(N)$$-optimal
86190.bm8 86190bm4 [1, 0, 1, 2514857562, -51791164403912] [4] 185794560

## Rank

sage: E.rank()

The elliptic curves in class 86190.bm have rank $$0$$.

## Modular form 86190.2.a.bm

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.