# Properties

 Label 1805.a Number of curves $2$ Conductor $1805$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 1805.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1805.a1 1805a2 $$[0, 1, 1, -105171, -9810339]$$ $$7575076864/1953125$$ $$33171021564453125$$ $$[]$$ $$14364$$ $$1.8793$$
1805.a2 1805a1 $$[0, 1, 1, -36581, 2679900]$$ $$318767104/125$$ $$2122945380125$$ $$$$ $$4788$$ $$1.3300$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1805.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1805.a do not have complex multiplication.

## Modular form1805.2.a.a

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 