Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*x - 1)
gp: K = bnfinit(y^23 - 7*y^22 + 2*y^21 + 88*y^20 - 150*y^19 - 422*y^18 + 1148*y^17 + 862*y^16 - 4167*y^15 - 105*y^14 + 8563*y^13 - 2935*y^12 - 10393*y^11 + 5920*y^10 + 7285*y^9 - 5405*y^8 - 2661*y^7 + 2503*y^6 + 354*y^5 - 537*y^4 + 21*y^3 + 39*y^2 - 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*x - 1)
\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} + \cdots - 1 \)
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: | $[11, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1487890299453836387819925171690895703966953\)
\(\medspace = 53\cdot 317\cdot 438479\cdot 20\!\cdots\!07\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(68.17\) | 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: | $53^{1/2}317^{1/2}438479^{1/2}201970054143124213916544634856807^{1/2}\approx 1.219791088446639e+21$ | ||
| Ramified primes: |
\(53\), \(317\), \(438479\), \(20197\!\cdots\!56807\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{14878\!\cdots\!66953}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $16$ | 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: |
$a$, $a^{22}-6a^{21}-4a^{20}+84a^{19}-66a^{18}-488a^{17}+660a^{16}+1522a^{15}-2645a^{14}-2750a^{13}+5813a^{12}+2878a^{11}-7515a^{10}-1595a^{9}+5690a^{8}+285a^{7}-2376a^{6}+127a^{5}+481a^{4}-56a^{3}-35a^{2}+4a+2$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7285a^{8}-5405a^{7}-2661a^{6}+2503a^{5}+354a^{4}-537a^{3}+21a^{2}+38a-2$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7284a^{8}-5402a^{7}-2656a^{6}+2485a^{5}+348a^{4}-505a^{3}+20a^{2}+23a$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7285a^{8}-5404a^{7}-2662a^{6}+2496a^{5}+359a^{4}-522a^{3}+13a^{2}+28a+1$, $3a^{22}-23a^{21}+20a^{20}+260a^{19}-625a^{18}-971a^{17}+4280a^{16}+361a^{15}-14228a^{14}+7599a^{13}+26136a^{12}-24592a^{11}-26377a^{10}+36063a^{9}+12272a^{8}-28099a^{7}+254a^{6}+11240a^{5}-2373a^{4}-1891a^{3}+664a^{2}+46a-26$, $6a^{22}-43a^{21}+20a^{20}+520a^{19}-991a^{18}-2303a^{17}+7236a^{16}+3607a^{15}-25207a^{14}+4640a^{13}+49071a^{12}-27590a^{11}-54640a^{10}+46310a^{9}+32396a^{8}-38570a^{7}-7218a^{6}+16236a^{5}-1356a^{4}-2895a^{3}+719a^{2}+87a-30$, $2a^{22}-13a^{21}-3a^{20}+178a^{19}-213a^{18}-988a^{17}+1876a^{16}+2799a^{15}-7397a^{14}-4028a^{13}+16445a^{12}+1877a^{11}-21866a^{10}+2353a^{9}+17316a^{8}-3775a^{7}-7704a^{6}+1964a^{5}+1658a^{4}-376a^{3}-112a^{2}+10a+1$, $8a^{22}-53a^{21}-a^{20}+685a^{19}-946a^{18}-3487a^{17}+7576a^{16}+8457a^{15}-27699a^{14}-7848a^{13}+56489a^{12}-6692a^{11}-67043a^{10}+23734a^{9}+44767a^{8}-23565a^{7}-14535a^{6}+10736a^{5}+1115a^{4}-2001a^{3}+302a^{2}+54a-12$, $4a^{22}-29a^{21}+16a^{20}+342a^{19}-678a^{18}-1459a^{17}+4780a^{16}+2104a^{15}-16115a^{14}+3191a^{13}+30327a^{12}-16144a^{11}-32904a^{10}+24369a^{9}+20087a^{8}-17728a^{7}-6516a^{6}+6244a^{5}+1137a^{4}-945a^{3}-138a^{2}+55a+11$, $8a^{22}-64a^{21}+71a^{20}+694a^{19}-1899a^{18}-2266a^{17}+12623a^{16}-1721a^{15}-40964a^{14}+30550a^{13}+72624a^{12}-88040a^{11}-67799a^{10}+125729a^{9}+23063a^{8}-98212a^{7}+10914a^{6}+40173a^{5}-11279a^{4}-7036a^{3}+2752a^{2}+184a-107$, $2a^{22}-15a^{21}+10a^{20}+179a^{19}-379a^{18}-770a^{17}+2714a^{16}+1062a^{15}-9441a^{14}+2254a^{13}+18518a^{12}-10862a^{11}-20995a^{10}+17639a^{9}+12972a^{8}-14525a^{7}-3372a^{6}+6088a^{5}-245a^{4}-1085a^{3}+226a^{2}+33a-9$, $14a^{22}-95a^{21}+9a^{20}+1224a^{19}-1831a^{18}-6189a^{17}+14509a^{16}+14749a^{15}-53603a^{14}-12589a^{13}+112240a^{12}-15091a^{11}-140402a^{10}+46320a^{9}+104539a^{8}-44851a^{7}-44116a^{6}+20359a^{5}+9451a^{4}-4084a^{3}-837a^{2}+249a+44$, $a^{6}-a^{5}-5a^{4}+4a^{3}+5a^{2}-4a$, $12a^{22}-81a^{21}+4a^{20}+1055a^{19}-1534a^{18}-5426a^{17}+12360a^{16}+13366a^{15}-46271a^{14}-12894a^{13}+98279a^{12}-9691a^{11}-124943a^{10}+37169a^{9}+94633a^{8}-37638a^{7}-40446a^{6}+17337a^{5}+8564a^{4}-3342a^{3}-668a^{2}+148a+23$, $11a^{22}-80a^{21}+43a^{20}+960a^{19}-1903a^{18}-4182a^{17}+13747a^{16}+6128a^{15}-47626a^{14}+10381a^{13}+92282a^{12}-54282a^{11}-102217a^{10}+88883a^{9}+60173a^{8}-72933a^{7}-13210a^{6}+30306a^{5}-2557a^{4}-5347a^{3}+1325a^{2}+164a-56$
| 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: | \( 5355593277510 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{6}\cdot 5355593277510 \cdot 1}{2\cdot\sqrt{1487890299453836387819925171690895703966953}}\cr\approx \mathstrut & 0.276631287710710 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*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 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*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 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*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 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 21*x^3 + 39*x^2 - 2*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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | $17{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | R | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(53\)
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 53.21.0.1 | $x^{21} - 5 x + 20$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
|
\(317\)
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(438479\)
| 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(201\!\cdots\!807\)
| 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |