# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("400.d1")

sage: E.isogeny_class()

## Elliptic curves in class 400.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400.d1 400c2 [0, -1, 0, -2008, 35312] [] 144
400.d2 400c3 [0, -1, 0, -1208, -19088] [] 240
400.d3 400c1 [0, -1, 0, -8, 112] [] 48 $$\Gamma_0(N)$$-optimal
400.d4 400c4 [0, -1, 0, 8792, 140912] [] 720

## Rank

sage: E.rank()

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

## Modular form400.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 