# Properties

 Label 75810bu Number of curves $6$ Conductor $75810$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("75810.bv1")

sage: E.isogeny_class()

## Elliptic curves in class 75810bu

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.bv6 75810bu1 [1, 0, 1, 3602, 116768] [2] 221184 $$\Gamma_0(N)$$-optimal
75810.bv5 75810bu2 [1, 0, 1, -25278, 1202656] [2, 2] 442368
75810.bv4 75810bu3 [1, 0, 1, -133578, -17771504] [2] 884736
75810.bv2 75810bu4 [1, 0, 1, -379058, 89789168] [2, 2] 884736
75810.bv3 75810bu5 [1, 0, 1, -353788, 102282656] [2] 1769472
75810.bv1 75810bu6 [1, 0, 1, -6064808, 5748247568] [2] 1769472

## Rank

sage: E.rank()

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

## Modular form 75810.2.a.bv

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.