# Properties

 Label 209814db Number of curves 6 Conductor 209814 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("209814.t1")

sage: E.isogeny_class()

## Elliptic curves in class 209814db

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
209814.t5 209814db1 [1, 1, 0, -1189674, -463696812] [2] 5898240 $$\Gamma_0(N)$$-optimal
209814.t4 209814db2 [1, 1, 0, -3987194, 2525733060] [2, 2] 11796480
209814.t2 209814db3 [1, 1, 0, -60636974, 181708987200] [2, 2] 23592960
209814.t6 209814db4 [1, 1, 0, 7902266, 14721941128] [2] 23592960
209814.t1 209814db5 [1, 1, 0, -970180664, 11630863048182] [2] 47185920
209814.t3 209814db6 [1, 1, 0, -57489764, 201416186778] [2] 47185920

## Rank

sage: E.rank()

The elliptic curves in class 209814db have rank $$0$$.

## Modular form 209814.2.a.t

sage: E.q_eigenform(10)

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

## 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.