# Properties

 Label 58800fd Number of curves $8$ Conductor $58800$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("58800.cq1")

sage: E.isogeny_class()

## Elliptic curves in class 58800fd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
58800.cq7 58800fd1 [0, -1, 0, -804008, -97369488] [2] 1327104 $$\Gamma_0(N)$$-optimal
58800.cq5 58800fd2 [0, -1, 0, -7076008, 7178150512] [2, 2] 2654208
58800.cq4 58800fd3 [0, -1, 0, -52548008, -146598745488] [2] 3981312
58800.cq6 58800fd4 [0, -1, 0, -1588008, 18022438512] [2] 5308416
58800.cq2 58800fd5 [0, -1, 0, -112916008, 461866790512] [2] 5308416
58800.cq3 58800fd6 [0, -1, 0, -52940008, -144300057488] [2, 2] 7962624
58800.cq8 58800fd7 [0, -1, 0, 14287992, -485818297488] [2] 15925248
58800.cq1 58800fd8 [0, -1, 0, -126440008, 344327942512] [2] 15925248

## Rank

sage: E.rank()

The elliptic curves in class 58800fd have rank $$1$$.

## Modular form 58800.2.a.cq

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} + 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.