Properties

Label 20.0.103...021.1
Degree $20$
Signature $(0, 10)$
Discriminant $1.037\times 10^{42}$
Root discriminant \(126.12\)
Ramified primes see page
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $S_{20}$ (as 20T1117)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x + 7)
 
Copy content gp:K = bnfinit(y^20 - 7*y + 7, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x + 7);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 7*x + 7)
 

\( x^{20} - 7x + 7 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1037398203776905299351207857106906772728021\) \(\medspace = 3\cdot 7^{19}\cdot 59\cdot 1986351023\cdot 258853157374157\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(126.12\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}7^{19/20}59^{1/2}1986351023^{1/2}258853157374157^{1/2}\approx 60587705215401.875$
Ramified primes:   \(3\), \(7\), \(59\), \(1986351023\), \(258853157374157\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  $\Q(\sqrt{63706\!\cdots\!25029}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a-1$, $7a^{19}-16a^{18}-7a^{17}-12a^{16}-39a^{15}-21a^{14}-5a^{13}-36a^{12}-3a^{11}+35a^{10}-5a^{9}+20a^{8}+81a^{7}+8a^{6}+4a^{5}+85a^{4}-22a^{3}-70a^{2}+54a-106$, $25a^{19}-5a^{18}+15a^{17}-21a^{16}-2a^{15}-26a^{14}-2a^{13}+7a^{12}+15a^{11}+35a^{10}-11a^{9}+37a^{8}-71a^{7}+32a^{6}-115a^{5}+100a^{4}-103a^{3}+184a^{2}-115a+29$, $12a^{19}+13a^{18}+6a^{17}-10a^{16}-24a^{15}-15a^{14}+16a^{13}+41a^{12}+27a^{11}-30a^{10}-71a^{9}-27a^{8}+65a^{7}+89a^{6}+8a^{5}-90a^{4}-91a^{3}+12a^{2}+95a-13$, $4a^{19}-13a^{18}-8a^{17}-44a^{16}+26a^{15}-29a^{14}+63a^{13}-61a^{12}+25a^{11}-89a^{10}+49a^{9}-23a^{8}+86a^{7}-61a^{6}+16a^{5}-129a^{4}+79a^{3}-23a^{2}+134a-55$, $10a^{19}-5a^{18}-33a^{17}+34a^{16}-32a^{15}+43a^{14}+13a^{13}-28a^{12}-6a^{11}-61a^{10}+93a^{9}-62a^{8}+87a^{7}-21a^{6}-119a^{5}+114a^{4}-148a^{3}+212a^{2}-87a-27$, $10a^{19}+191a^{18}-158a^{17}+107a^{16}+33a^{15}-314a^{14}+260a^{13}-162a^{12}-108a^{11}+520a^{10}-433a^{9}+263a^{8}+302a^{7}-870a^{6}+700a^{5}-363a^{4}-642a^{3}+1433a^{2}-1131a+442$, $261a^{19}+35a^{18}-161a^{17}+208a^{16}-25a^{15}-399a^{14}+177a^{13}-49a^{12}-588a^{11}+32a^{10}+166a^{9}-966a^{8}+30a^{7}+364a^{6}-1214a^{5}-200a^{4}+851a^{3}-1489a^{2}-566a-337$, $604a^{19}-224a^{18}+871a^{17}-350a^{16}+685a^{15}+169a^{14}-175a^{13}+1136a^{12}-1269a^{11}+1601a^{10}-1117a^{9}+827a^{8}+468a^{7}-1642a^{6}+2823a^{5}-3588a^{4}+3415a^{3}-2305a^{2}+500a-1952$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 6395590809330 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 6395590809330 \cdot 2}{2\cdot\sqrt{1037398203776905299351207857106906772728021}}\cr\approx \mathstrut & 0.602152657647239 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x + 7) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 7*x + 7, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x + 7); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 7*x + 7); 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

$S_{20}$ (as 20T1117):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 2432902008176640000
The 627 conjugacy class representatives for $S_{20}$
Character table for $S_{20}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 40 sibling: 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 ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ R ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R $19{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $18{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ $17{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $15{,}\,{\href{/padicField/47.5.0.1}{5} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ R

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.18.1.0a1.1$x^{18} + x^{10} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{4} + 2 x^{2} + 2$$1$$18$$0$$C_{18}$$$[\ ]^{18}$$
\(7\) Copy content Toggle raw display 7.1.20.19a1.1$x^{20} + 7$$20$$1$$19$20T18$$[\ ]_{20}^{4}$$
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$$[\ ]$$
59.1.2.1a1.2$x^{2} + 118$$2$$1$$1$$C_2$$$[\ ]_{2}$$
59.8.1.0a1.1$x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$$1$$8$$0$$C_8$$$[\ ]^{8}$$
59.8.1.0a1.1$x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(1986351023\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $12$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$
\(258853157374157\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $15$$1$$15$$0$$C_{15}$$$[\ ]^{15}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)