# Properties

 Label 67830.s Number of curves 8 Conductor 67830 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("67830.s1")
sage: E.isogeny_class()

## Elliptic curves in class 67830.s

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
67830.s1 67830p8 [1, 0, 1, -2903932529, 60196943047556] 2 59719680
67830.s2 67830p5 [1, 0, 1, -2903465114, 60217301097812] 6 19906560
67830.s3 67830p6 [1, 0, 1, -217399409, 541937811332] 4 29859840
67830.s4 67830p2 [1, 0, 1, -181466594, 940883727476] 12 9953280
67830.s5 67830p4 [1, 0, 1, -180445994, 951990304916] 6 19906560
67830.s6 67830p3 [1, 0, 1, -111231089, -445724835964] 2 14929920
67830.s7 67830p1 [1, 0, 1, -11405474, 14526794612] 6 4976640 $$\Gamma_0(N)$$-optimal
67830.s8 67830p7 [1, 0, 1, 770440591, 4097766675332] 2 59719680

## Rank

sage: E.rank()

The elliptic curves in class 67830.s have rank $$1$$.

## Modular form 67830.2.a.s

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.