# Properties

 Label 75810l Number of curves $8$ Conductor $75810$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("75810.l1")

sage: E.isogeny_class()

## Elliptic curves in class 75810l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.l7 75810l1 [1, 1, 0, -14808, 237888] [2] 331776 $$\Gamma_0(N)$$-optimal
75810.l5 75810l2 [1, 1, 0, -130328, -17991168] [2, 2] 663552
75810.l4 75810l3 [1, 1, 0, -967848, 366083952] [2] 995328
75810.l6 75810l4 [1, 1, 0, -29248, -45060392] [2] 1327104
75810.l2 75810l5 [1, 1, 0, -2079728, -1155271128] [2] 1327104
75810.l3 75810l6 [1, 1, 0, -975068, 360335388] [2, 2] 1990656
75810.l8 75810l7 [1, 1, 0, 263162, 1214466442] [2] 3981312
75810.l1 75810l8 [1, 1, 0, -2328818, -861559362] [2] 3981312

## Rank

sage: E.rank()

The elliptic curves in class 75810l have rank $$1$$.

## Modular form 75810.2.a.l

sage: E.q_eigenform(10)

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

## 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 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.