# Properties

 Label 2005.b Number of curves 4 Conductor 2005 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2005.b1")

sage: E.isogeny_class()

## Elliptic curves in class 2005.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2005.b1 2005a3 [1, -1, 1, -53467, -4745166]  1792
2005.b2 2005a2 [1, -1, 1, -3342, -73516] [2, 2] 896
2005.b3 2005a4 [1, -1, 1, -3217, -79366]  1792
2005.b4 2005a1 [1, -1, 1, -217, -1016]  448 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 2005.b have rank $$1$$.

## Modular form2005.2.a.b

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels. 