# Properties

 Label 55470.bd Number of curves $8$ Conductor $55470$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 55470.bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.bd1 55470bc7 [1, 1, 1, -9861680, 11915842025] [2] 1935360
55470.bd2 55470bc8 [1, 1, 1, -838560, 39877737] [2] 1935360
55470.bd3 55470bc6 [1, 1, 1, -616680, 185786025] [2, 2] 967680
55470.bd4 55470bc5 [1, 1, 1, -533475, -150195765] [2] 645120
55470.bd5 55470bc4 [1, 1, 1, -126695, 14897747] [2] 645120
55470.bd6 55470bc2 [1, 1, 1, -34245, -2223993] [2, 2] 322560
55470.bd7 55470bc3 [1, 1, 1, -25000, 4968617] [2] 483840
55470.bd8 55470bc1 [1, 1, 1, 2735, -167905] [2] 161280 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 55470.bd have rank $$0$$.

## Modular form 55470.2.a.bd

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + 2q^{13} + 4q^{14} - q^{15} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.