# Properties

 Label 33810ca Number of curves $6$ Conductor $33810$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("33810.cd1")

sage: E.isogeny_class()

## Elliptic curves in class 33810ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33810.cd5 33810ca1 [1, 1, 1, -20581, -1255381] [2] 147456 $$\Gamma_0(N)$$-optimal
33810.cd4 33810ca2 [1, 1, 1, -338101, -75809077] [2, 2] 294912
33810.cd3 33810ca3 [1, 1, 1, -346921, -71656621] [2, 2] 589824
33810.cd1 33810ca4 [1, 1, 1, -5409601, -4845047677] [2] 589824
33810.cd6 33810ca5 [1, 1, 1, 430709, -346315537] [2] 1179648
33810.cd2 33810ca6 [1, 1, 1, -1265671, 468935879] [2] 1179648

## Rank

sage: E.rank()

The elliptic curves in class 33810ca have rank $$1$$.

## Modular form 33810.2.a.cd

sage: E.q_eigenform(10)

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

## 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.