# Properties

 Label 21.a Number of curves $6$ Conductor $21$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 21.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21.a1 21a5 $$[1, 0, 0, -784, -8515]$$ $$53297461115137/147$$ $$147$$ $$$$ $$4$$ $$0.074205$$
21.a2 21a2 $$[1, 0, 0, -49, -136]$$ $$13027640977/21609$$ $$21609$$ $$[2, 2]$$ $$2$$ $$-0.27237$$
21.a3 21a3 $$[1, 0, 0, -39, 90]$$ $$6570725617/45927$$ $$45927$$ $$$$ $$2$$ $$-0.27237$$
21.a4 21a6 $$[1, 0, 0, -34, -217]$$ $$-4354703137/17294403$$ $$-17294403$$ $$$$ $$4$$ $$0.074205$$
21.a5 21a1 $$[1, 0, 0, -4, -1]$$ $$7189057/3969$$ $$3969$$ $$[2, 4]$$ $$1$$ $$-0.61894$$ $$\Gamma_0(N)$$-optimal
21.a6 21a4 $$[1, 0, 0, 1, 0]$$ $$103823/63$$ $$-63$$ $$$$ $$2$$ $$-0.96552$$

## Rank

sage: E.rank()

The elliptic curves in class 21.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 21.a do not have complex multiplication.

## Modular form21.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - 2q^{15} - q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 