# Properties

 Label 400e Number of curves 4 Conductor 400 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("400.c1")

sage: E.isogeny_class()

## Elliptic curves in class 400e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400.c3 400e1 [0, 1, 0, -33, -62]  48 $$\Gamma_0(N)$$-optimal
400.c4 400e2 [0, 1, 0, 92, -312]  96
400.c1 400e3 [0, 1, 0, -1033, 12438]  144
400.c2 400e4 [0, 1, 0, -908, 15688]  288

## Rank

sage: E.rank()

The elliptic curves in class 400e have rank $$0$$.

## Modular form400.2.a.c

sage: E.q_eigenform(10)

$$q - 2q^{3} + 2q^{7} + q^{9} - 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 