# Properties

 Label 400a Number of curves 4 Conductor 400 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("400.e1")

sage: E.isogeny_class()

## Elliptic curves in class 400a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400.e3 400a1 [0, 0, 0, -50, -125]  48 $$\Gamma_0(N)$$-optimal
400.e2 400a2 [0, 0, 0, -175, 750] [2, 2] 96
400.e1 400a3 [0, 0, 0, -2675, 53250]  192
400.e4 400a4 [0, 0, 0, 325, 4250]  192

## Rank

sage: E.rank()

The elliptic curves in class 400a have rank $$1$$.

## Modular form400.2.a.e

sage: E.q_eigenform(10)

$$q - 4q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 