# Properties

 Label 40425.cm Number of curves $6$ Conductor $40425$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("40425.cm1")

sage: E.isogeny_class()

## Elliptic curves in class 40425.cm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40425.cm1 40425w6 [1, 1, 0, -16231275, 25161675750] [2] 2359296
40425.cm2 40425w4 [1, 1, 0, -1071900, 345778875] [2, 2] 1179648
40425.cm3 40425w2 [1, 1, 0, -330775, -68510000] [2, 2] 589824
40425.cm4 40425w1 [1, 1, 0, -324650, -71333625] [2] 294912 $$\Gamma_0(N)$$-optimal
40425.cm5 40425w3 [1, 1, 0, 312350, -301964375] [2] 1179648
40425.cm6 40425w5 [1, 1, 0, 2229475, 2059192500] [2] 2359296

## Rank

sage: E.rank()

The elliptic curves in class 40425.cm have rank $$1$$.

## Modular form 40425.2.a.cm

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + q^{11} + q^{12} - 2q^{13} - q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.