# Properties

 Label 254898.de Number of curves $6$ Conductor $254898$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("254898.de1")

sage: E.isogeny_class()

## Elliptic curves in class 254898.de

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
254898.de1 254898de6 [1, -1, 0, -1748422363851, -889851231810145571] [2] 3397386240
254898.de2 254898de4 [1, -1, 0, -109484301411, -13848343276727723] [2, 2] 1698693120
254898.de3 254898de5 [1, -1, 0, -37292089851, -31838079298229555] [2] 3397386240
254898.de4 254898de2 [1, -1, 0, -11562685731, 120273121639957] [2, 2] 849346560
254898.de5 254898de1 [1, -1, 0, -8952530211, 325631283424789] [2] 424673280 $$\Gamma_0(N)$$-optimal
254898.de6 254898de3 [1, -1, 0, 44596441629, 945935843912149] [2] 1698693120

## Rank

sage: E.rank()

The elliptic curves in class 254898.de have rank $$1$$.

## Modular form 254898.2.a.de

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.