# Properties

 Label 83490bb Number of curves $6$ Conductor $83490$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("83490.bc1")

sage: E.isogeny_class()

## Elliptic curves in class 83490bb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
83490.bc5 83490bb1 [1, 0, 1, -50823, -4842422] [2] 491520 $$\Gamma_0(N)$$-optimal
83490.bc4 83490bb2 [1, 0, 1, -834903, -293697494] [2, 2] 983040
83490.bc3 83490bb3 [1, 0, 1, -856683, -277571582] [2, 2] 1966080
83490.bc1 83490bb4 [1, 0, 1, -13358403, -18793411694] [2] 1966080
83490.bc6 83490bb5 [1, 0, 1, 1063587, -1344473594] [2] 3932160
83490.bc2 83490bb6 [1, 0, 1, -3125433, 1821475918] [2] 3932160

## Rank

sage: E.rank()

The elliptic curves in class 83490bb have rank $$0$$.

## Modular form 83490.2.a.bc

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.