# Properties

 Label 86490r Number of curves $6$ Conductor $86490$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("86490.f1")

sage: E.isogeny_class()

## Elliptic curves in class 86490r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86490.f6 86490r1 [1, -1, 0, 518760, -814942400] [2] 3932160 $$\Gamma_0(N)$$-optimal
86490.f5 86490r2 [1, -1, 0, -10551960, -12463553984] [2, 2] 7864320
86490.f4 86490r3 [1, -1, 0, -32001480, 54334541200] [2] 15728640
86490.f2 86490r4 [1, -1, 0, -166233960, -824905639184] [2, 2] 15728640
86490.f3 86490r5 [1, -1, 0, -163639260, -851904530564] [2] 31457280
86490.f1 86490r6 [1, -1, 0, -2659740660, -52796068384604] [2] 31457280

## Rank

sage: E.rank()

The elliptic curves in class 86490r have rank $$1$$.

## Modular form 86490.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4q^{11} - 6q^{13} + q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.