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Properties

Label 20.0.337...000.2
Degree $20$
Signature $[0, 10]$
Discriminant $3.371\times 10^{35}$
Root discriminant \(59.76\)
Ramified primes $2,5,3469$
Class number $3766$ (GRH)
Class group [3766] (GRH)
Galois group $C_2^8.D_5^2:C_2^2$ (as 20T756)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469)
 
gp: K = bnfinit(y^20 + 42*y^18 + 733*y^16 + 6936*y^14 + 39054*y^12 + 135432*y^10 + 289182*y^8 + 367352*y^6 + 254077*y^4 + 76318*y^2 + 3469, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469)
 

\( x^{20} + 42 x^{18} + 733 x^{16} + 6936 x^{14} + 39054 x^{12} + 135432 x^{10} + 289182 x^{8} + \cdots + 3469 \) 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:   \(337133094159738515866255360000000000\) \(\medspace = 2^{36}\cdot 5^{10}\cdot 3469^{5}\) 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:  \(59.76\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(3469\) 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:  \(\Q(\sqrt{3469}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{8}a^{16}-\frac{1}{2}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{17}-\frac{1}{2}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{3}{8}a$, $\frac{1}{3022848120136}a^{18}+\frac{133282889827}{3022848120136}a^{16}-\frac{79613553755}{755712030034}a^{14}+\frac{45449873909}{377856015017}a^{12}-\frac{42506953309}{377856015017}a^{10}+\frac{242372440157}{1511424060068}a^{8}+\frac{16267459603}{755712030034}a^{6}+\frac{52777911516}{377856015017}a^{4}+\frac{623877397635}{3022848120136}a^{2}+\frac{1275443906551}{3022848120136}$, $\frac{1}{3022848120136}a^{19}+\frac{133282889827}{3022848120136}a^{17}-\frac{79613553755}{755712030034}a^{15}+\frac{45449873909}{377856015017}a^{13}-\frac{42506953309}{377856015017}a^{11}+\frac{242372440157}{1511424060068}a^{9}+\frac{16267459603}{755712030034}a^{7}+\frac{52777911516}{377856015017}a^{5}+\frac{623877397635}{3022848120136}a^{3}+\frac{1275443906551}{3022848120136}a$ 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

$C_{3766}$, which has order $3766$ (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:   $\frac{1627}{2034188}a^{18}+\frac{123783}{4068376}a^{16}+\frac{235414}{508547}a^{14}+\frac{3665867}{1017094}a^{12}+\frac{7761694}{508547}a^{10}+\frac{34882659}{1017094}a^{8}+\frac{71896995}{2034188}a^{6}+\frac{14611261}{2034188}a^{4}-\frac{4034447}{508547}a^{2}-\frac{8411393}{4068376}$, $\frac{14659564555}{3022848120136}a^{18}+\frac{295527545731}{1511424060068}a^{16}+\frac{4875777941011}{1511424060068}a^{14}+\frac{10658750522773}{377856015017}a^{12}+\frac{214415969947161}{1511424060068}a^{10}+\frac{630452081219815}{1511424060068}a^{8}+\frac{262432322953961}{377856015017}a^{6}+\frac{897307389906583}{1511424060068}a^{4}+\frac{611786488296401}{3022848120136}a^{2}+\frac{11221863428323}{1511424060068}$, $\frac{268942147}{116263389236}a^{18}-\frac{21833329175}{232526778472}a^{16}-\frac{181516104431}{116263389236}a^{14}-\frac{1601013765679}{116263389236}a^{12}-\frac{8124090355909}{116263389236}a^{10}-\frac{24085649217843}{116263389236}a^{8}-\frac{20182328721453}{58131694618}a^{6}-\frac{8643233948859}{29065847309}a^{4}-\frac{11449473916049}{116263389236}a^{2}+\frac{7497501155}{232526778472}$, $\frac{2700448549}{3022848120136}a^{18}+\frac{25242819557}{755712030034}a^{16}+\frac{373717089087}{755712030034}a^{14}+\frac{5551135503463}{1511424060068}a^{12}+\frac{5357969574881}{377856015017}a^{10}+\frac{19194945736833}{755712030034}a^{8}+\frac{11261418696835}{1511424060068}a^{6}-\frac{12665026978873}{377856015017}a^{4}-\frac{102102097571915}{3022848120136}a^{2}-\frac{7614722879139}{1511424060068}$, $\frac{14659564555}{3022848120136}a^{18}-\frac{295527545731}{1511424060068}a^{16}-\frac{4875777941011}{1511424060068}a^{14}-\frac{10658750522773}{377856015017}a^{12}-\frac{214415969947161}{1511424060068}a^{10}-\frac{630452081219815}{1511424060068}a^{8}-\frac{262432322953961}{377856015017}a^{6}-\frac{897307389906583}{1511424060068}a^{4}-\frac{611786488296401}{3022848120136}a^{2}-\frac{9710439368255}{1511424060068}$, $\frac{1180491255}{3022848120136}a^{18}-\frac{5314690754}{377856015017}a^{16}-\frac{149551861473}{755712030034}a^{14}-\frac{516635592691}{377856015017}a^{12}-\frac{7165288889565}{1511424060068}a^{10}-\frac{2702210008465}{377856015017}a^{8}-\frac{2000587143037}{1511424060068}a^{6}+\frac{6783548008259}{1511424060068}a^{4}-\frac{2192839122653}{3022848120136}a^{2}-\frac{3255689204349}{1511424060068}$, $\frac{10182919065}{3022848120136}a^{18}+\frac{204854531491}{1511424060068}a^{16}+\frac{841795679580}{377856015017}a^{14}+\frac{14627381885465}{755712030034}a^{12}+\frac{145579532930815}{1511424060068}a^{10}+\frac{105295308180233}{377856015017}a^{8}+\frac{686158856974947}{1511424060068}a^{6}+\frac{573628354454867}{1511424060068}a^{4}+\frac{390583355254871}{3022848120136}a^{2}+\frac{4343492837677}{755712030034}$, $\frac{11704570545}{3022848120136}a^{18}+\frac{237642565917}{1511424060068}a^{16}+\frac{989203558349}{377856015017}a^{14}+\frac{8753857435950}{377856015017}a^{12}+\frac{178909032270695}{1511424060068}a^{10}+\frac{268743426113839}{755712030034}a^{8}+\frac{922969592885403}{1511424060068}a^{6}+\frac{829152950578533}{1511424060068}a^{4}+\frac{626927919182191}{3022848120136}a^{2}+\frac{10826471902683}{755712030034}$, $\frac{17116600733}{3022848120136}a^{18}-\frac{344951807139}{1511424060068}a^{16}-\frac{5689993483375}{1511424060068}a^{14}-\frac{12441105088858}{377856015017}a^{12}-\frac{62654726172809}{377856015017}a^{10}-\frac{370108673962505}{755712030034}a^{8}-\frac{311823907511982}{377856015017}a^{6}-\frac{10\!\cdots\!81}{1511424060068}a^{4}-\frac{791206935056061}{3022848120136}a^{2}-\frac{8879110461665}{755712030034}$ 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:  \( 828338.9338584389 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 828338.9338584389 \cdot 3766}{2\cdot\sqrt{337133094159738515866255360000000000}}\cr\approx \mathstrut & 0.257606243745277 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469)
 
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 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469, 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 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469);
 
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 + 42*x^18 + 733*x^16 + 6936*x^14 + 39054*x^12 + 135432*x^10 + 289182*x^8 + 367352*x^6 + 254077*x^4 + 76318*x^2 + 3469);
 
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

$C_2^8.D_5^2:C_2^2$ (as 20T756):

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 solvable group of order 102400
The 130 conjugacy class representatives for $C_2^8.D_5^2:C_2^2$
Character table for $C_2^8.D_5^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 Deg $20$$10$$2$$36$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(3469\) Copy content Toggle raw display $\Q_{3469}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{3469}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{3469}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{3469}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
Deg $4$$4$$1$$3$