# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("75810.cn1")

sage: E.isogeny_class()

## Elliptic curves in class 75810co

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.cn7 75810co1 [1, 1, 1, -179605, -29360773] [2] 580608 $$\Gamma_0(N)$$-optimal
75810.cn6 75810co2 [1, 1, 1, -208485, -19322085] [2, 2] 1161216
75810.cn5 75810co3 [1, 1, 1, -531580, 113151197] [2] 1741824
75810.cn8 75810co4 [1, 1, 1, 694015, -140979085] [2] 2322432
75810.cn4 75810co5 [1, 1, 1, -1573065, 745388547] [2] 2322432
75810.cn2 75810co6 [1, 1, 1, -7924860, 8582892765] [2, 2] 3483648
75810.cn3 75810co7 [1, 1, 1, -7347260, 9887806685] [2] 6967296
75810.cn1 75810co8 [1, 1, 1, -126794940, 549489304797] [2] 6967296

## Rank

sage: E.rank()

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

## Modular form 75810.2.a.cn

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.