# Properties

 Label 4005.d Number of curves 4 Conductor 4005 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4005.d1")

sage: E.isogeny_class()

## Elliptic curves in class 4005.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4005.d1 4005c3 [1, -1, 0, -21375, 1208196]  7168
4005.d2 4005c2 [1, -1, 0, -1350, 18711] [2, 2] 3584
4005.d3 4005c1 [1, -1, 0, -225, -864]  1792 $$\Gamma_0(N)$$-optimal
4005.d4 4005c4 [1, -1, 0, 675, 68526]  7168

## Rank

sage: E.rank()

The elliptic curves in class 4005.d have rank $$0$$.

## Modular form4005.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} - 4q^{11} - 6q^{13} - q^{16} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 