# Properties

 Label 1001.b Number of curves 4 Conductor 1001 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1001.b1")

sage: E.isogeny_class()

## Elliptic curves in class 1001.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1001.b1 1001b3 [1, -1, 1, -9916, -377564]  608
1001.b2 1001b4 [1, -1, 1, -1006, 2552]  608
1001.b3 1001b2 [1, -1, 1, -621, -5764] [2, 2] 304
1001.b4 1001b1 [1, -1, 1, -16, -198]  152 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1001.b have rank $$0$$.

## Modular form1001.2.a.b

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 