# Properties

 Label 405.b Number of curves $2$ Conductor $405$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 405.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405.b1 405d2 $$[1, -1, 1, -2027, 35776]$$ $$-15590912409/78125$$ $$-4613203125$$ $$[]$$ $$252$$ $$0.70167$$
405.b2 405d1 $$[1, -1, 1, -2, -26]$$ $$-9/5$$ $$-295245$$ $$[]$$ $$36$$ $$-0.27128$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 405.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405.b do not have complex multiplication.

## Modular form405.2.a.b

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 