# Properties

 Label 46410.ck Number of curves $8$ Conductor $46410$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("ck1")

E.isogeny_class()

## Elliptic curves in class 46410.ck

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
46410.ck1 46410cn8 $$[1, 0, 0, -2258527940, -41313148043070]$$ $$1274090022584975661628188489514561/14072533302105480763470$$ $$14072533302105480763470$$ $$[2]$$ $$25165824$$ $$3.8194$$
46410.ck2 46410cn6 $$[1, 0, 0, -141270590, -644445412800]$$ $$311802066473807207098058600161/1033693082103011001480900$$ $$1033693082103011001480900$$ $$[2, 2]$$ $$12582912$$ $$3.4728$$
46410.ck3 46410cn4 $$[1, 0, 0, -139015370, 630861993012]$$ $$297106512928238351998640242081/3977028808593750000$$ $$3977028808593750000$$ $$[8]$$ $$6291456$$ $$3.1262$$
46410.ck4 46410cn7 $$[1, 0, 0, -80405240, -1203420614130]$$ $$-57487943130312093140621093761/592356094985924086700006670$$ $$-592356094985924086700006670$$ $$[2]$$ $$25165824$$ $$3.8194$$
46410.ck5 46410cn3 $$[1, 0, 0, -12746090, -254913900]$$ $$229010110533436633465952161/132501160769452503210000$$ $$132501160769452503210000$$ $$[2, 4]$$ $$6291456$$ $$3.1262$$
46410.ck6 46410cn2 $$[1, 0, 0, -8696090, 9838496100]$$ $$72727020009972527154752161/265361167808100000000$$ $$265361167808100000000$$ $$[2, 8]$$ $$3145728$$ $$2.7796$$
46410.ck7 46410cn1 $$[1, 0, 0, -298010, 293238372]$$ $$-2926956820564562516641/35459588343029760000$$ $$-35459588343029760000$$ $$[8]$$ $$1572864$$ $$2.4331$$ $$\Gamma_0(N)$$-optimal
46410.ck8 46410cn5 $$[1, 0, 0, 50978410, -2026455000]$$ $$14651516183052242700771495839/8480668142378708755560900$$ $$-8480668142378708755560900$$ $$[4]$$ $$12582912$$ $$3.4728$$

## Rank

sage: E.rank()

The elliptic curves in class 46410.ck have rank $$0$$.

## Complex multiplication

The elliptic curves in class 46410.ck do not have complex multiplication.

## Modular form46410.2.a.ck

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.