# Properties

 Label 336336id Number of curves $6$ Conductor $336336$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("336336.id1")

sage: E.isogeny_class()

## Elliptic curves in class 336336id

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
336336.id5 336336id1 [0, 1, 0, -18832, 1420628] [2] 1572864 $$\Gamma_0(N)$$-optimal
336336.id4 336336id2 [0, 1, 0, -336352, 74958260] [2, 2] 3145728
336336.id1 336336id3 [0, 1, 0, -5381392, 4803169748] [2] 6291456
336336.id3 336336id4 [0, 1, 0, -371632, 58235540] [2, 2] 6291456
336336.id6 336336id5 [0, 1, 0, 1051328, 398038388] [2] 12582912
336336.id2 336336id6 [0, 1, 0, -2359072, -1351256908] [2] 12582912

## Rank

sage: E.rank()

The elliptic curves in class 336336id have rank $$1$$.

## Modular form 336336.2.a.id

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} + q^{11} - q^{13} + 2q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.