# Properties

 Label 231.a Number of curves 6 Conductor 231 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("231.a1")

sage: E.isogeny_class()

## Elliptic curves in class 231.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
231.a1 231a5 [1, 1, 1, -4519, -118810]  160
231.a2 231a3 [1, 1, 1, -284, -1924] [2, 2] 80
231.a3 231a2 [1, 1, 1, -39, 36] [2, 4] 40
231.a4 231a1 [1, 1, 1, -34, 62]  20 $$\Gamma_0(N)$$-optimal
231.a5 231a6 [1, 1, 1, 31, -5578]  160
231.a6 231a4 [1, 1, 1, 126, 432]  80

## Rank

sage: E.rank()

The elliptic curves in class 231.a have rank $$0$$.

## Modular form231.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 