↑

Properties

Label 20.0.460...936.1
Degree $20$
Signature $[0, 10]$
Discriminant $4.608\times 10^{36}$
Root discriminant \(68.10\)
Ramified primes $2,41$
Class number $964$ (GRH)
Class group [2, 482] (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41)
 
gp: K = bnfinit(y^20 + 41*y^18 + 656*y^16 + 5289*y^14 + 23165*y^12 + 55801*y^10 + 71545*y^8 + 45141*y^6 + 13120*y^4 + 1517*y^2 + 41, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41)
 

\( x^{20} + 41 x^{18} + 656 x^{16} + 5289 x^{14} + 23165 x^{12} + 55801 x^{10} + 71545 x^{8} + 45141 x^{6} + \cdots + 41 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4607795443446634940843146923591335936\) \(\medspace = 2^{20}\cdot 41^{19}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(68.10\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 41^{19/20}\approx 68.104319772682$
Ramified primes:   \(2\), \(41\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{41}) \)
$\card{ \Gal(K/\Q) }$:  $20$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(164=2^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(131,·)$, $\chi_{164}(133,·)$, $\chi_{164}(81,·)$, $\chi_{164}(141,·)$, $\chi_{164}(143,·)$, $\chi_{164}(43,·)$, $\chi_{164}(87,·)$, $\chi_{164}(25,·)$, $\chi_{164}(91,·)$, $\chi_{164}(159,·)$, $\chi_{164}(155,·)$, $\chi_{164}(37,·)$, $\chi_{164}(39,·)$, $\chi_{164}(105,·)$, $\chi_{164}(103,·)$, $\chi_{164}(45,·)$, $\chi_{164}(113,·)$, $\chi_{164}(115,·)$, $\chi_{164}(57,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{7}$, $\frac{1}{657}a^{16}+\frac{2}{73}a^{14}+\frac{35}{219}a^{12}-\frac{10}{219}a^{10}+\frac{53}{657}a^{8}-\frac{34}{219}a^{4}-\frac{85}{219}a^{2}-\frac{110}{657}$, $\frac{1}{657}a^{17}+\frac{2}{73}a^{15}+\frac{35}{219}a^{13}-\frac{10}{219}a^{11}+\frac{53}{657}a^{9}-\frac{34}{219}a^{5}-\frac{85}{219}a^{3}-\frac{110}{657}a$, $\frac{1}{3858449967}a^{18}+\frac{1762651}{3858449967}a^{16}+\frac{93209570}{1286149989}a^{14}-\frac{9177052}{428716663}a^{12}-\frac{3830878}{52855479}a^{10}+\frac{338193440}{3858449967}a^{8}-\frac{562305310}{1286149989}a^{6}+\frac{75059731}{428716663}a^{4}-\frac{1443614141}{3858449967}a^{2}+\frac{653515540}{3858449967}$, $\frac{1}{3858449967}a^{19}+\frac{1762651}{3858449967}a^{17}+\frac{93209570}{1286149989}a^{15}-\frac{9177052}{428716663}a^{13}-\frac{3830878}{52855479}a^{11}+\frac{338193440}{3858449967}a^{9}-\frac{562305310}{1286149989}a^{7}+\frac{75059731}{428716663}a^{5}-\frac{1443614141}{3858449967}a^{3}+\frac{653515540}{3858449967}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{482}$, which has order $964$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{5859137}{1286149989}a^{18}+\frac{234505856}{1286149989}a^{16}+\frac{1204913977}{428716663}a^{14}+\frac{9155134917}{428716663}a^{12}+\frac{109027268116}{1286149989}a^{10}+\frac{221399449813}{1286149989}a^{8}+\frac{68423920823}{428716663}a^{6}+\frac{20880308616}{428716663}a^{4}+\frac{7498090067}{1286149989}a^{2}+\frac{518049746}{1286149989}$, $\frac{21956099}{3858449967}a^{18}+\frac{301328062}{1286149989}a^{16}+\frac{4847405720}{1286149989}a^{14}+\frac{39319525547}{1286149989}a^{12}+\frac{518744119510}{3858449967}a^{10}+\frac{413846386202}{1286149989}a^{8}+\frac{510242335241}{1286149989}a^{6}+\frac{284203309037}{1286149989}a^{4}+\frac{192805805255}{3858449967}a^{2}+\frac{4439541691}{1286149989}$, $\frac{46279708}{3858449967}a^{18}+\frac{1879550698}{3858449967}a^{16}+\frac{9875335246}{1286149989}a^{14}+\frac{77702485639}{1286149989}a^{12}+\frac{979439712233}{3858449967}a^{10}+\frac{2190260916182}{3858449967}a^{8}+\frac{810383758339}{1286149989}a^{6}+\frac{372852276115}{1286149989}a^{4}+\frac{175072577857}{3858449967}a^{2}+\frac{7300099609}{3858449967}$, $\frac{17256502}{1286149989}a^{18}+\frac{2107488046}{3858449967}a^{16}+\frac{11116532386}{1286149989}a^{14}+\frac{29353333202}{428716663}a^{12}+\frac{374473731610}{1286149989}a^{10}+\frac{2568799631885}{3858449967}a^{8}+\frac{992629011172}{1286149989}a^{6}+\frac{166556895878}{428716663}a^{4}+\frac{28048809922}{428716663}a^{2}-\frac{1516515344}{3858449967}$, $\frac{7153619}{3858449967}a^{18}+\frac{289180166}{3858449967}a^{16}+\frac{1515166744}{1286149989}a^{14}+\frac{11981887709}{1286149989}a^{12}+\frac{155401521877}{3858449967}a^{10}+\frac{380743380484}{3858449967}a^{8}+\frac{178515778366}{1286149989}a^{6}+\frac{137354054531}{1286149989}a^{4}+\frac{130635317930}{3858449967}a^{2}+\frac{5914403714}{3858449967}$, $\frac{5687290}{3858449967}a^{18}+\frac{231223946}{3858449967}a^{16}+\frac{1219993658}{1286149989}a^{14}+\frac{9712838225}{1286149989}a^{12}+\frac{126029508281}{3858449967}a^{10}+\frac{301007293507}{3858449967}a^{8}+\frac{127903887665}{1286149989}a^{6}+\frac{78283807058}{1286149989}a^{4}+\frac{51131145076}{3858449967}a^{2}-\frac{451091668}{3858449967}$, $\frac{11287574}{1286149989}a^{18}+\frac{452598058}{1286149989}a^{16}+\frac{2334756838}{428716663}a^{14}+\frac{53692411265}{1286149989}a^{12}+\frac{217549196014}{1286149989}a^{10}+\frac{464518008218}{1286149989}a^{8}+\frac{164138779737}{428716663}a^{6}+\frac{226454659475}{1286149989}a^{4}+\frac{41576248949}{1286149989}a^{2}+\frac{1795224907}{1286149989}$, $\frac{31629688}{3858449967}a^{18}+\frac{427087634}{1286149989}a^{16}+\frac{6702748136}{1286149989}a^{14}+\frac{17442245920}{428716663}a^{12}+\frac{649275504452}{3858449967}a^{10}+\frac{155724310168}{428716663}a^{8}+\frac{472343839040}{1286149989}a^{6}+\frac{50915056711}{428716663}a^{4}-\frac{31454656244}{3858449967}a^{2}-\frac{5142337436}{1286149989}$, $\frac{26709103}{3858449967}a^{18}+\frac{1068614008}{3858449967}a^{16}+\frac{5489103770}{1286149989}a^{14}+\frac{13903507852}{428716663}a^{12}+\frac{497246066165}{3858449967}a^{10}+\frac{1012810644170}{3858449967}a^{8}+\frac{313900560845}{1286149989}a^{6}+\frac{29700201648}{428716663}a^{4}-\frac{5906571083}{3858449967}a^{2}-\frac{4387178960}{3858449967}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5104264.63655 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 5104264.63655 \cdot 964}{2\cdot\sqrt{4607795443446634940843146923591335936}}\cr\approx \mathstrut & 0.109908760252 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{20}$ (as 20T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.1102736.1, 5.5.2825761.1, 10.10.327381934393961.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{5}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ $20$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ R ${\href{/padicField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.10.10.11$x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$$2$$5$$10$$C_{10}$$[2]^{5}$
\(41\) Copy content Toggle raw display 41.20.19.1$x^{20} + 41$$20$$1$$19$20T1$[\ ]_{20}$