# Properties

 Label 105.a Number of curves $4$ Conductor $105$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 105.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
105.a1 105a3 $$[1, 0, 1, -113, -469]$$ $$157551496201/13125$$ $$13125$$ $$$$ $$16$$ $$-0.16596$$
105.a2 105a2 $$[1, 0, 1, -8, -7]$$ $$47045881/11025$$ $$11025$$ $$[2, 2]$$ $$8$$ $$-0.51254$$
105.a3 105a1 $$[1, 0, 1, -3, 1]$$ $$1771561/105$$ $$105$$ $$$$ $$4$$ $$-0.85911$$ $$\Gamma_0(N)$$-optimal
105.a4 105a4 $$[1, 0, 1, 17, -37]$$ $$590589719/972405$$ $$-972405$$ $$$$ $$16$$ $$-0.16596$$

## Rank

sage: E.rank()

The elliptic curves in class 105.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 105.a do not have complex multiplication.

## Modular form105.2.a.a

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 