# Properties

 Label 47040.cs Number of curves 4 Conductor 47040 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.cs1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.cs1 47040fb4 [0, -1, 0, -352865, -80555775]  393216
47040.cs2 47040fb2 [0, -1, 0, -23585, -1067583] [2, 2] 196608
47040.cs3 47040fb1 [0, -1, 0, -7905, 258945]  98304 $$\Gamma_0(N)$$-optimal
47040.cs4 47040fb3 [0, -1, 0, 54815, -6728063]  393216

## Rank

sage: E.rank()

The elliptic curves in class 47040.cs have rank $$0$$.

## Modular form 47040.2.a.cs

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 6q^{13} - q^{15} - 2q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 