# Properties

 Label 47040dm Number of curves 8 Conductor 47040 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.ha1")

sage: E.isogeny_class()

## Elliptic curves in class 47040dm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.ha7 47040dm1 [0, 1, 0, 658495, -154761825] [2] 1179648 $$\Gamma_0(N)$$-optimal
47040.ha6 47040dm2 [0, 1, 0, -3355585, -1388690017] [2, 2] 2359296
47040.ha5 47040dm3 [0, 1, 0, -23676865, 43346575775] [2, 2] 4718592
47040.ha4 47040dm4 [0, 1, 0, -47259585, -125031134817] [2] 4718592
47040.ha8 47040dm5 [0, 1, 0, 3982655, 138589366943] [2] 9437184
47040.ha2 47040dm6 [0, 1, 0, -376476865, 2811485935775] [2, 2] 9437184
47040.ha3 47040dm7 [0, 1, 0, -374124865, 2848350713375] [2] 18874368
47040.ha1 47040dm8 [0, 1, 0, -6023628865, 179941184998175] [2] 18874368

## Rank

sage: E.rank()

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

## Modular form 47040.2.a.ha

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 4q^{11} - 2q^{13} + q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.