# Properties

 Label 110670.cm Number of curves $8$ Conductor $110670$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("cm1")

E.isogeny_class()

## Elliptic curves in class 110670.cm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
110670.cm1 110670cp8 $$[1, 0, 0, -258731208280, -50654853221670100]$$ $$1915447099311696788795300773853656437121/2359330710565906457438184052500$$ $$2359330710565906457438184052500$$ $$[2]$$ $$788529152$$ $$5.1081$$
110670.cm2 110670cp4 $$[1, 0, 0, -36426360100, 2675908664482832]$$ $$5345287166085790635663218704920974401/226978257155929261920000$$ $$226978257155929261920000$$ $$[8]$$ $$197132288$$ $$4.4149$$
110670.cm3 110670cp6 $$[1, 0, 0, -16306945780, -777467273017600]$$ $$479558500651862026155270257138637121/16398477508430925116750756250000$$ $$16398477508430925116750756250000$$ $$[2, 2]$$ $$394264576$$ $$4.7615$$
110670.cm4 110670cp3 $$[1, 0, 0, -2525695780, 32101233232400]$$ $$1781832302709884421209712638637121/585975791707934414062500000000$$ $$585975791707934414062500000000$$ $$[2, 4]$$ $$197132288$$ $$4.4149$$
110670.cm5 110670cp2 $$[1, 0, 0, -2276760100, 41806588162832]$$ $$1305195379419707692723460338574401/268915325631261926400000000$$ $$268915325631261926400000000$$ $$[2, 8]$$ $$98566144$$ $$4.0683$$
110670.cm6 110670cp1 $$[1, 0, 0, -126851620, 800523760400]$$ $$-225741686871429146260559062081/146211902909299839467520000$$ $$-146211902909299839467520000$$ $$[8]$$ $$49283072$$ $$3.7218$$ $$\Gamma_0(N)$$-optimal
110670.cm7 110670cp7 $$[1, 0, 0, 5617316720, -2712410599365100]$$ $$19602454118850102896203267379162879/3189643093950949672061474484052500$$ $$-3189643093950949672061474484052500$$ $$[2]$$ $$788529152$$ $$5.1081$$
110670.cm8 110670cp5 $$[1, 0, 0, 7272583340, 220528019677472]$$ $$42539250356844378683161410804461759/45626792684197425842285156250000$$ $$-45626792684197425842285156250000$$ $$[4]$$ $$394264576$$ $$4.7615$$

## Rank

sage: E.rank()

The elliptic curves in class 110670.cm have rank $$0$$.

## Complex multiplication

The elliptic curves in class 110670.cm do not have complex multiplication.

## Modular form 110670.2.a.cm

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 4 & 8 \\ 16 & 1 & 8 & 4 & 2 & 4 & 16 & 8 \\ 2 & 8 & 1 & 2 & 4 & 8 & 2 & 4 \\ 4 & 4 & 2 & 1 & 2 & 4 & 4 & 2 \\ 8 & 2 & 4 & 2 & 1 & 2 & 8 & 4 \\ 16 & 4 & 8 & 4 & 2 & 1 & 16 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 1 & 8 \\ 8 & 8 & 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.