# Properties

 Label 11a Number of curves $3$ Conductor $11$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 11a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
11.a2 11a1 $$[0, -1, 1, -10, -20]$$ $$-122023936/161051$$ $$-161051$$ $$$$ $$1$$ $$-0.30801$$ $$\Gamma_0(N)$$-optimal
11.a1 11a2 $$[0, -1, 1, -7820, -263580]$$ $$-52893159101157376/11$$ $$-11$$ $$[]$$ $$5$$ $$0.49671$$
11.a3 11a3 $$[0, -1, 1, 0, 0]$$ $$-4096/11$$ $$-11$$ $$$$ $$5$$ $$-1.1127$$

## Rank

sage: E.rank()

The elliptic curves in class 11a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 11a do not have complex multiplication.

## Modular form11.2.a.a

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrr} 1 & 5 & 5 \\ 5 & 1 & 25 \\ 5 & 25 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 