# Properties

 Label 427119.i Number of curves 6 Conductor 427119 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("427119.i1")

sage: E.isogeny_class()

## Elliptic curves in class 427119.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
427119.i1 427119i5 [1, 0, 1, -8355670, 9295810493] [2] 12902400 $$\Gamma_0(N)$$-optimal*
427119.i2 427119i3 [1, 0, 1, -525155, 143504561] [2, 2] 6451200 $$\Gamma_0(N)$$-optimal*
427119.i3 427119i2 [1, 0, 1, -72150, -4175069] [2, 2] 3225600 $$\Gamma_0(N)$$-optimal*
427119.i4 427119i1 [1, 0, 1, -62905, -6075841] [2] 1612800 $$\Gamma_0(N)$$-optimal*
427119.i5 427119i6 [1, 0, 1, 57280, 444506969] [2] 12902400
427119.i6 427119i4 [1, 0, 1, 232935, -30168311] [2] 6451200
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 427119.i4.

## Rank

sage: E.rank()

The elliptic curves in class 427119.i have rank $$1$$.

## Modular form 427119.2.a.i

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.