# Properties

 Label 82110.bs Number of curves $8$ Conductor $82110$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("bs1")

E.isogeny_class()

## Elliptic curves in class 82110.bs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82110.bs1 82110bt8 $$[1, 0, 0, -25879815780, -1602472070375340]$$ $$1916934412547006969354120058646317121/161260899317970551066940$$ $$161260899317970551066940$$ $$[2]$$ $$117440512$$ $$4.3407$$
82110.bs2 82110bt6 $$[1, 0, 0, -1617601080, -25035066997200]$$ $$468099305477291219804418298216321/135739171637240733141123600$$ $$135739171637240733141123600$$ $$[2, 2]$$ $$58720256$$ $$3.9941$$
82110.bs3 82110bt7 $$[1, 0, 0, -1409434380, -31714678433460]$$ $$-309640881349964101700603056547521/255321054523588348291733959740$$ $$-255321054523588348291733959740$$ $$[2]$$ $$117440512$$ $$4.3407$$
82110.bs4 82110bt4 $$[1, 0, 0, -114223080, -283150929600]$$ $$164810665209657549410090824321/60743509039087274201760000$$ $$60743509039087274201760000$$ $$[2, 4]$$ $$29360128$$ $$3.6475$$
82110.bs5 82110bt2 $$[1, 0, 0, -49423080, 130545230400]$$ $$13350979617415439280823624321/363628103290905600000000$$ $$363628103290905600000000$$ $$[2, 8]$$ $$14680064$$ $$3.3010$$
82110.bs6 82110bt1 $$[1, 0, 0, -49095400, 132402324032]$$ $$13087181362848921857775657601/10116926999101440000$$ $$10116926999101440000$$ $$[8]$$ $$7340032$$ $$2.9544$$ $$\Gamma_0(N)$$-optimal
82110.bs7 82110bt3 $$[1, 0, 0, 10134040, 425388708672]$$ $$115099000398621243971890559/78239096930039062500000000$$ $$-78239096930039062500000000$$ $$[8]$$ $$29360128$$ $$3.6475$$
82110.bs8 82110bt5 $$[1, 0, 0, 352354920, -2007529902000]$$ $$4837987390362436347081585367679/4540848316592979232425603600$$ $$-4540848316592979232425603600$$ $$[4]$$ $$58720256$$ $$3.9941$$

## Rank

sage: E.rank()

The elliptic curves in class 82110.bs have rank $$1$$.

## Complex multiplication

The elliptic curves in class 82110.bs do not have complex multiplication.

## Modular form 82110.2.a.bs

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.