# Properties

 Label 363a Number of curves 4 Conductor 363 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("363.b1")
sage: E.isogeny_class()

## Elliptic curves in class 363a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
363.b3 363a1 [1, 1, 1, -789, 8130] 4 180 $$\Gamma_0(N)$$-optimal
363.b2 363a2 [1, 1, 1, -1394, -6874] 4 360
363.b1 363a3 [1, 1, 1, -17729, -915100] 2 720
363.b4 363a4 [1, 1, 1, 5261, -46804] 2 720

## Rank

sage: E.rank()

The elliptic curves in class 363a have rank $$0$$.

## Modular form363.2.a.b

sage: E.q_eigenform(10)
$$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 4q^{7} + 3q^{8} + q^{9} + 2q^{10} + q^{12} + 2q^{13} + 4q^{14} + 2q^{15} - q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 