# Properties

 Label 86190r Number of curves 4 Conductor 86190 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("86190.t1")

sage: E.isogeny_class()

## Elliptic curves in class 86190r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86190.t3 86190r1 [1, 1, 0, -17072, 742656]  442368 $$\Gamma_0(N)$$-optimal
86190.t2 86190r2 [1, 1, 0, -71152, -6579776] [2, 2] 884736
86190.t4 86190r3 [1, 1, 0, 97848, -32909976]  1769472
86190.t1 86190r4 [1, 1, 0, -1105432, -447803624]  1769472

## Rank

sage: E.rank()

The elliptic curves in class 86190r have rank $$1$$.

## Modular form 86190.2.a.t

sage: E.q_eigenform(10)

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

## 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 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. 