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 + 297*x^4 + 89*x^3 - 27*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 + 297*y^4 + 89*y^3 - 27*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 + 297*x^4 + 89*x^3 - 27*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 + 297*x^4 + 89*x^3 - 27*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: |
\(-696771975062209357134867192571475966589863\)
\(\medspace = -\,17\cdot 40\!\cdots\!39\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.96\) | 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: | $17^{1/2}40986586768365256302051011327733880387639^{1/2}\approx 8.347286835027352e+20$ | ||
| Ramified primes: |
\(17\), \(40986\!\cdots\!87639\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-69677\!\cdots\!89863}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8160a^{9}-4433a^{8}+4919a^{7}+2383a^{6}-1652a^{5}-677a^{4}+297a^{3}+89a^{2}-27a-4$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8160a^{9}-4433a^{8}+4919a^{7}+2383a^{6}-1652a^{5}-677a^{4}+297a^{3}+88a^{2}-26a-2$, $a^{3}-3a-1$, $2a^{22}-5a^{21}-39a^{20}+92a^{19}+337a^{18}-717a^{17}-1705a^{16}+3078a^{15}+5586a^{14}-7914a^{13}-12275a^{12}+12374a^{11}+18010a^{10}-11338a^{9}-16999a^{8}+5427a^{7}+9598a^{6}-948a^{5}-2882a^{4}-64a^{3}+398a^{2}+25a-18$, $a^{21}-3a^{20}-18a^{19}+55a^{18}+141a^{17}-429a^{16}-638a^{15}+1858a^{14}+1864a^{13}-4889a^{12}-3693a^{11}+8033a^{10}+4989a^{9}-8159a^{8}-4423a^{7}+4911a^{6}+2348a^{5}-1630a^{4}-626a^{3}+273a^{2}+61a-19$, $22a^{22}-62a^{21}-406a^{20}+1132a^{19}+3287a^{18}-8767a^{17}-15487a^{16}+37518a^{15}+47288a^{14}-96764a^{13}-97586a^{12}+153671a^{11}+135851a^{10}-146983a^{9}-122409a^{8}+79040a^{7}+65610a^{6}-20937a^{5}-18234a^{4}+2293a^{3}+2280a^{2}-62a-92$, $2a^{22}-6a^{21}-36a^{20}+110a^{19}+282a^{18}-858a^{17}-1276a^{16}+3716a^{15}+3728a^{14}-9778a^{13}-7385a^{12}+16064a^{11}+9969a^{10}-16300a^{9}-8819a^{8}+9765a^{7}+4667a^{6}-3185a^{5}-1248a^{4}+510a^{3}+126a^{2}-32a-1$, $8a^{22}-26a^{21}-137a^{20}+472a^{19}+1003a^{18}-3642a^{17}-4157a^{16}+15586a^{15}+10939a^{14}-40457a^{13}-19439a^{12}+65369a^{11}+23887a^{10}-64829a^{9}-19719a^{8}+37440a^{7}+9863a^{6}-11400a^{5}-2451a^{4}+1575a^{3}+254a^{2}-69a-7$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8160a^{9}-4433a^{8}+4919a^{7}+2383a^{6}-1652a^{5}-676a^{4}+295a^{3}+86a^{2}-22a-3$, $a^{22}-2a^{21}-21a^{20}+37a^{19}+196a^{18}-288a^{17}-1067a^{16}+1220a^{15}+3722a^{14}-3025a^{13}-8582a^{12}+4340a^{11}+13023a^{10}-3170a^{9}-12592a^{8}+486a^{7}+7292a^{6}+729a^{5}-2296a^{4}-369a^{3}+346a^{2}+46a-19$, $4a^{21}-12a^{20}-71a^{19}+217a^{18}+546a^{17}-1661a^{16}-2412a^{15}+7006a^{14}+6832a^{13}-17741a^{12}-12992a^{11}+27497a^{10}+16555a^{9}-25406a^{8}-13370a^{7}+12955a^{6}+5992a^{5}-3167a^{4}-1115a^{3}+345a^{2}+59a-16$, $3a^{22}-8a^{21}-57a^{20}+147a^{19}+478a^{18}-1146a^{17}-2343a^{16}+4936a^{15}+7450a^{14}-12803a^{13}-15968a^{12}+20406a^{11}+23003a^{10}-19491a^{9}-21456a^{8}+10331a^{7}+12044a^{6}-2588a^{5}-3620a^{4}+226a^{3}+502a^{2}+3a-22$, $16a^{22}-49a^{21}-283a^{20}+891a^{19}+2168a^{18}-6878a^{17}-9550a^{16}+29386a^{15}+27085a^{14}-75890a^{13}-52091a^{12}+121302a^{11}+68425a^{10}-117855a^{9}-58939a^{8}+65528a^{7}+30203a^{6}-18611a^{5}-7774a^{4}+2307a^{3}+855a^{2}-84a-27$, $5a^{22}-14a^{21}-93a^{20}+257a^{19}+760a^{18}-2004a^{17}-3619a^{16}+8652a^{15}+11178a^{14}-22581a^{13}-23354a^{12}+36472a^{11}+32982a^{10}-35809a^{9}-30314a^{8}+20154a^{7}+16790a^{6}-5856a^{5}-4960a^{4}+786a^{3}+685a^{2}-37a-33$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3693a^{12}+8033a^{11}+4990a^{10}-8160a^{9}-4433a^{8}+4919a^{7}+2383a^{6}-1652a^{5}-676a^{4}+296a^{3}+85a^{2}-24a-1$, $5a^{22}-15a^{21}-89a^{20}+272a^{19}+688a^{18}-2093a^{17}-3065a^{16}+8910a^{15}+8791a^{14}-22918a^{13}-17022a^{12}+36473a^{11}+22295a^{10}-35276a^{9}-18895a^{8}+19528a^{7}+9374a^{6}-5536a^{5}-2276a^{4}+699a^{3}+225a^{2}-27a-7$, $a^{21}-4a^{20}-15a^{19}+72a^{18}+89a^{17}-553a^{16}-260a^{15}+2370a^{14}+368a^{13}-6211a^{12}-193a^{11}+10221a^{10}+68a^{9}-10377a^{8}-360a^{7}+6080a^{6}+468a^{5}-1793a^{4}-161a^{3}+222a^{2}+17a-8$, $7a^{22}-22a^{21}-122a^{20}+400a^{19}+914a^{18}-3089a^{17}-3896a^{16}+13215a^{15}+10553a^{14}-34231a^{13}-19113a^{12}+55063a^{11}+23296a^{10}-54232a^{9}-18178a^{8}+31134a^{7}+7871a^{6}-9638a^{5}-1264a^{4}+1525a^{3}-43a^{2}-106a+17$, $a^{22}-3a^{21}-18a^{20}+55a^{19}+141a^{18}-429a^{17}-638a^{16}+1858a^{15}+1864a^{14}-4889a^{13}-3694a^{12}+8034a^{11}+5002a^{10}-8168a^{9}-4488a^{8}+4937a^{7}+2501a^{6}-1654a^{5}-792a^{4}+271a^{3}+130a^{2}-13a-7$
| 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: | \( 36539551739300 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{3}\cdot 36539551739300 \cdot 1}{2\cdot\sqrt{696771975062209357134867192571475966589863}}\cr\approx \mathstrut & 0.711602415533196 \end{aligned}\] (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 + 297*x^4 + 89*x^3 - 27*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 + 297*x^4 + 89*x^3 - 27*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 + 297*x^4 + 89*x^3 - 27*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 + 297*x^4 + 89*x^3 - 27*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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $23$ | $15{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $18{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.9.0.1 | $x^{9} + 7 x^{2} + 8 x + 14$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(409\!\cdots\!639\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |