# Properties

 Label 117670.m Number of curves $4$ Conductor $117670$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("117670.m1")

sage: E.isogeny_class()

## Elliptic curves in class 117670.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
117670.m1 117670m4 [1, -1, 1, -449983, -116064619]  1126400
117670.m2 117670m3 [1, -1, 1, -147403, 20392237]  1126400
117670.m3 117670m2 [1, -1, 1, -29733, -1588519] [2, 2] 563200
117670.m4 117670m1 [1, -1, 1, 3887, -149583]  281600 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 117670.m have rank $$1$$.

## Modular form 117670.2.a.m

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3q^{9} - q^{10} - 4q^{11} + 6q^{13} + q^{14} + q^{16} - 2q^{17} - 3q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 