# Properties

 Label 116886l Number of curves $6$ Conductor $116886$ CM no Rank $1$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("l1")

sage: E.isogeny_class()

## Elliptic curves in class 116886l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116886.m6 116886l1 $$[1, 0, 1, 12610133, -2212140394]$$ $$125177609053596564863/73635189229502208$$ $$-130449229466606161106688$$ $$[2]$$ $$12288000$$ $$3.1248$$ $$\Gamma_0(N)$$-optimal
116886.m5 116886l2 $$[1, 0, 1, -50900347, -17784910090]$$ $$8232463578739844255617/4687062591766850064$$ $$8303417292133072666229904$$ $$[2, 2]$$ $$24576000$$ $$3.4714$$
116886.m3 116886l3 $$[1, 0, 1, -521544367, 4563463980590]$$ $$8856076866003496152467137/46664863048067576004$$ $$82669651446297643013222244$$ $$[2, 2]$$ $$49152000$$ $$3.8180$$
116886.m2 116886l4 $$[1, 0, 1, -596424007, -5595437019394]$$ $$13244420128496241770842177/29965867631164664892$$ $$53086362426533704900736412$$ $$[2]$$ $$49152000$$ $$3.8180$$
116886.m4 116886l5 $$[1, 0, 1, -239264677, 9484050624794]$$ $$-855073332201294509246497/21439133060285771735058$$ $$-37980732003412922060731085538$$ $$[2]$$ $$98304000$$ $$4.1645$$
116886.m1 116886l6 $$[1, 0, 1, -8334128377, 292844688915986]$$ $$36136672427711016379227705697/1011258101510224722$$ $$1791505413569555218731042$$ $$[2]$$ $$98304000$$ $$4.1645$$

## Rank

sage: E.rank()

The elliptic curves in class 116886l have rank $$1$$.

## Complex multiplication

The elliptic curves in class 116886l do not have complex multiplication.

## Modular form 116886.2.a.l

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.