# Properties

 Label 21780.q Number of curves 4 Conductor 21780 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("21780.q1")

sage: E.isogeny_class()

## Elliptic curves in class 21780.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21780.q1 21780y3 [0, 0, 0, -45012, -3674891]  51840
21780.q2 21780y4 [0, 0, 0, -39567, -4597274]  103680
21780.q3 21780y1 [0, 0, 0, -1452, 14641]  17280 $$\Gamma_0(N)$$-optimal
21780.q4 21780y2 [0, 0, 0, 3993, 98494]  34560

## Rank

sage: E.rank()

The elliptic curves in class 21780.q have rank $$1$$.

## Modular form 21780.2.a.q

sage: E.q_eigenform(10)

$$q + q^{5} - 2q^{7} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 