# Properties

 Label 14400dw Number of curves 4 Conductor 14400 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("14400.dz1")

sage: E.isogeny_class()

## Elliptic curves in class 14400dw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
14400.dz3 14400dw1 [0, 0, 0, -1200, 11000]  9216 $$\Gamma_0(N)$$-optimal
14400.dz4 14400dw2 [0, 0, 0, 3300, 74000]  18432
14400.dz1 14400dw3 [0, 0, 0, -37200, -2761000]  27648
14400.dz2 14400dw4 [0, 0, 0, -32700, -3454000]  55296

## Rank

sage: E.rank()

The elliptic curves in class 14400dw have rank $$1$$.

## Modular form 14400.2.a.dz

sage: E.q_eigenform(10)

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

## 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. 