Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1)
gp: K = bnfinit(y^23 - 3*y^22 - 18*y^21 + 55*y^20 + 141*y^19 - 429*y^18 - 638*y^17 + 1858*y^16 + 1864*y^15 - 4889*y^14 - 3693*y^13 + 8033*y^12 + 4990*y^11 - 8160*y^10 - 4433*y^9 + 4919*y^8 + 2383*y^7 - 1652*y^6 - 677*y^5 + 298*y^4 + 89*y^3 - 28*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1)
\( x^{23} - 3 x^{22} - 18 x^{21} + 55 x^{20} + 141 x^{19} - 429 x^{18} - 638 x^{17} + 1858 x^{16} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[17, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1141341323415075008135606599043792165942047\)
\(\medspace = -\,1223\cdot 879051344165227\cdot 1061634057189152640819707\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(67.39\) | 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: | $1223^{1/2}879051344165227^{1/2}1061634057189152640819707^{1/2}\approx 1.0683357727863816e+21$ | ||
| Ramified primes: |
\(1223\), \(879051344165227\), \(1061634057189152640819707\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-11413\!\cdots\!42047}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$a$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8160a^{9}-4433a^{8}+4919a^{7}+2383a^{6}-1652a^{5}-677a^{4}+298a^{3}+88a^{2}-28a-1$, $10a^{22}-27a^{21}-187a^{20}+490a^{19}+1540a^{18}-3760a^{17}-7402a^{16}+15861a^{15}+23060a^{14}-39975a^{13}-48354a^{12}+61062a^{11}+67795a^{10}-54373a^{9}-60679a^{8}+25084a^{7}+31600a^{6}-4238a^{5}-8165a^{4}-128a^{3}+878a^{2}+47a-30$, $9a^{22}-23a^{21}-173a^{20}+421a^{19}+1468a^{18}-3259a^{17}-7265a^{16}+13866a^{15}+23191a^{14}-35217a^{13}-49418a^{12}+54093a^{11}+69760a^{10}-48181a^{9}-62364a^{8}+21880a^{7}+32241a^{6}-3331a^{5}-8203a^{4}-244a^{3}+842a^{2}+44a-26$, $12a^{22}-35a^{21}-218a^{20}+639a^{19}+1729a^{18}-4952a^{17}-7944a^{16}+21229a^{15}+23589a^{14}-54948a^{13}-47355a^{12}+87840a^{11}+64300a^{10}-85030a^{9}-56608a^{8}+46777a^{7}+29501a^{6}-12992a^{5}-7807a^{4}+1586a^{3}+905a^{2}-69a-36$, $8a^{22}-20a^{21}-155a^{20}+366a^{19}+1327a^{18}-2830a^{17}-6627a^{16}+12008a^{15}+21327a^{14}-30329a^{13}-45724a^{12}+46073a^{11}+64761a^{10}-40087a^{9}-57906a^{8}+17124a^{7}+29844a^{6}-1876a^{5}-7557a^{4}-441a^{3}+785a^{2}+58a-26$, $6a^{22}-16a^{21}-113a^{20}+291a^{19}+939a^{18}-2241a^{17}-4558a^{16}+9507a^{15}+14325a^{14}-24174a^{13}-30205a^{12}+37459a^{11}+42401a^{10}-34222a^{9}-37869a^{8}+16714a^{7}+19694a^{6}-3462a^{5}-5146a^{4}+166a^{3}+580a^{2}+5a-22$, $2a^{22}-6a^{21}-36a^{20}+110a^{19}+282a^{18}-858a^{17}-1276a^{16}+3716a^{15}+3728a^{14}-9778a^{13}-7386a^{12}+16066a^{11}+9980a^{10}-16320a^{9}-8866a^{8}+9838a^{7}+4765a^{6}-3302a^{5}-1348a^{4}+585a^{3}+169a^{2}-42a-6$, $75a^{22}-198a^{21}-1422a^{20}+3616a^{19}+11886a^{18}-27943a^{17}-57952a^{16}+118801a^{15}+182639a^{14}-302058a^{13}-385667a^{12}+466051a^{11}+541704a^{10}-420111a^{9}-483293a^{8}+197368a^{7}+249534a^{6}-35167a^{5}-63356a^{4}-173a^{3}+6600a^{2}+237a-214$, $3a^{22}-2a^{21}-75a^{20}+40a^{19}+805a^{18}-318a^{17}-4862a^{16}+1249a^{15}+18169a^{14}-2256a^{13}-43446a^{12}+98a^{11}+66338a^{10}+6838a^{9}-62521a^{8}-11535a^{7}+33779a^{6}+7743a^{5}-9183a^{4}-1934a^{3}+1076a^{2}+129a-42$, $123a^{22}-325a^{21}-2330a^{20}+5931a^{19}+19459a^{18}-45796a^{17}-94803a^{16}+194536a^{15}+298578a^{14}-494147a^{13}-630069a^{12}+761589a^{11}+884235a^{10}-685562a^{9}-787883a^{8}+321408a^{7}+405998a^{6}-57001a^{5}-102761a^{4}-343a^{3}+10653a^{2}+388a-343$, $a^{22}-3a^{21}-17a^{20}+51a^{19}+127a^{18}-361a^{17}-563a^{16}+1372a^{15}+1680a^{14}-2986a^{13}-3524a^{12}+3575a^{11}+5052a^{10}-1764a^{9}-4531a^{8}-588a^{7}+2169a^{6}+995a^{5}-391a^{4}-305a^{3}+2a^{2}+24a+2$, $31a^{22}-83a^{21}-585a^{20}+1517a^{19}+4864a^{18}-11742a^{17}-23591a^{16}+50074a^{15}+74038a^{14}-128014a^{13}-156046a^{12}+199486a^{11}+219547a^{10}-183305a^{9}-197233a^{8}+89888a^{7}+103461a^{6}-18338a^{5}-27185a^{4}+592a^{3}+3004a^{2}+82a-101$, $47a^{22}-124a^{21}-891a^{20}+2263a^{19}+7449a^{18}-17475a^{17}-36340a^{16}+74240a^{15}+114634a^{14}-188604a^{13}-242331a^{12}+290704a^{11}+340733a^{10}-261629a^{9}-304264a^{8}+122469a^{7}+157237a^{6}-21492a^{5}-39974a^{4}-253a^{3}+4162a^{2}+160a-134$, $6a^{22}-17a^{21}-111a^{20}+312a^{19}+900a^{18}-2431a^{17}-4238a^{16}+10476a^{15}+12886a^{14}-27232a^{13}-26326a^{12}+43623a^{11}+35935a^{10}-42121a^{9}-31190a^{8}+22917a^{7}+15487a^{6}-6234a^{5}-3588a^{4}+791a^{3}+276a^{2}-53a+3$, $28a^{22}-74a^{21}-530a^{20}+1348a^{19}+4427a^{18}-10388a^{17}-21598a^{16}+44027a^{15}+68201a^{14}-111512a^{13}-144419a^{12}+171128a^{11}+203407a^{10}-152849a^{9}-181799a^{8}+70350a^{7}+93887a^{6}-11584a^{5}-23799a^{4}-405a^{3}+2472a^{2}+117a-80$, $a^{22}-4a^{21}-16a^{20}+75a^{19}+106a^{18}-605a^{17}-385a^{16}+2749a^{15}+897a^{14}-7720a^{13}-1635a^{12}+13784a^{11}+2709a^{10}-15432a^{9}-3556a^{8}+10236a^{7}+2827a^{6}-3603a^{5}-1066a^{4}+580a^{3}+151a^{2}-33a-3$, $56a^{22}-147a^{21}-1063a^{20}+2680a^{19}+8903a^{18}-20666a^{17}-43530a^{16}+87620a^{15}+137641a^{14}-221919a^{13}-291579a^{12}+340352a^{11}+410547a^{10}-303482a^{9}-366710a^{8}+139020a^{7}+189280a^{6}-22407a^{5}-47968a^{4}-977a^{3}+4981a^{2}+243a-164$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8159a^{9}-4433a^{8}+4909a^{7}+2381a^{6}-1619a^{5}-666a^{4}+258a^{3}+73a^{2}-16a-1$
| 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: | \( 50170522197400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{3}\cdot 50170522197400 \cdot 1}{2\cdot\sqrt{1141341323415075008135606599043792165942047}}\cr\approx \mathstrut & 0.763414467934608 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1)
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^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1, 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^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1);
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^23 - 3*x^22 - 18*x^21 + 55*x^20 + 141*x^19 - 429*x^18 - 638*x^17 + 1858*x^16 + 1864*x^15 - 4889*x^14 - 3693*x^13 + 8033*x^12 + 4990*x^11 - 8160*x^10 - 4433*x^9 + 4919*x^8 + 2383*x^7 - 1652*x^6 - 677*x^5 + 298*x^4 + 89*x^3 - 28*x^2 - 4*x + 1);
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
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 non-solvable group of order 25852016738884976640000 |
| The 1255 conjugacy class representatives for $S_{23}$ are not computed |
| Character table for $S_{23}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 46 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 | $19{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/23.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | $20{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $23$ | $20{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.7.0.1}{7} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(1223\)
| $\Q_{1223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
|
\(879051344165227\)
| $\Q_{879051344165227}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(106\!\cdots\!707\)
| $\Q_{10\!\cdots\!07}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |