# Properties

 Label 21021b Number of curves $6$ Conductor $21021$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("21021.e1")

sage: E.isogeny_class()

## Elliptic curves in class 21021b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21021.e5 21021b1 [1, 1, 1, -1177, -22786] [2] 24576 $$\Gamma_0(N)$$-optimal
21021.e4 21021b2 [1, 1, 1, -21022, -1181734] [2, 2] 49152
21021.e3 21021b3 [1, 1, 1, -23227, -921544] [2, 2] 98304
21021.e1 21021b4 [1, 1, 1, -336337, -75217696] [2] 98304
21021.e6 21021b5 [1, 1, 1, 65708, -6186496] [2] 196608
21021.e2 21021b6 [1, 1, 1, -147442, 21039668] [2] 196608

## Rank

sage: E.rank()

The elliptic curves in class 21021b have rank $$0$$.

## Modular form 21021.2.a.e

sage: E.q_eigenform(10)

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

## 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.