# Properties

 Label 88445bn Number of curves $3$ Conductor $88445$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bn1")

sage: E.isogeny_class()

## Elliptic curves in class 88445bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
88445.w2 88445bn1 $$[0, 1, 1, -23585, 1538779]$$ $$-262144/35$$ $$-193721529881915$$ $$[]$$ $$228096$$ $$1.4740$$ $$\Gamma_0(N)$$-optimal
88445.w3 88445bn2 $$[0, 1, 1, 153305, -3891744]$$ $$71991296/42875$$ $$-237308874105345875$$ $$[]$$ $$684288$$ $$2.0233$$
88445.w1 88445bn3 $$[0, 1, 1, -2323155, -1426494191]$$ $$-250523582464/13671875$$ $$-75672472610123046875$$ $$[]$$ $$2052864$$ $$2.5726$$

## Rank

sage: E.rank()

The elliptic curves in class 88445bn have rank $$2$$.

## Complex multiplication

The elliptic curves in class 88445bn do not have complex multiplication.

## Modular form 88445.2.a.bn

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} + q^{5} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} + q^{15} + 4q^{16} - 3q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.