Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*x^2 - 2*x + 1)
gp: K = bnfinit(y^23 - y^22 - 4*y^21 - 3*y^20 + 10*y^19 + 17*y^18 - 2*y^17 - 26*y^16 - 21*y^15 + 15*y^14 + 19*y^13 - 7*y^12 - 18*y^11 + 15*y^10 + 35*y^9 - 33*y^7 - 24*y^6 + 12*y^5 + 16*y^4 + 3*y^3 - 5*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*x^2 - 2*x + 1)
\( x^{23} - x^{22} - 4 x^{21} - 3 x^{20} + 10 x^{19} + 17 x^{18} - 2 x^{17} - 26 x^{16} - 21 x^{15} + \cdots + 1 \)
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: | $[9, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-9554598217771454170076503812873306955636\)
\(\medspace = -\,2^{2}\cdot 19\cdot 432841815513398891\cdot 290448827022730856621\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.74\) | 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: | $2\cdot 19^{1/2}432841815513398891^{1/2}290448827022730856621^{1/2}\approx 9.774762512599196e+19$ | ||
| Ramified primes: |
\(2\), \(19\), \(432841815513398891\), \(290448827022730856621\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-23886\!\cdots\!38909}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: |
$a^{22}-a^{21}-4a^{20}-3a^{19}+10a^{18}+17a^{17}-2a^{16}-26a^{15}-21a^{14}+15a^{13}+19a^{12}-7a^{11}-18a^{10}+15a^{9}+35a^{8}-33a^{6}-24a^{5}+12a^{4}+16a^{3}+3a^{2}-5a-2$, $4a^{22}-a^{21}-18a^{20}-23a^{19}+25a^{18}+88a^{17}+46a^{16}-77a^{15}-133a^{14}-23a^{13}+64a^{12}+2a^{11}-69a^{10}+18a^{9}+161a^{8}+98a^{7}-73a^{6}-137a^{5}-36a^{4}+40a^{3}+30a^{2}+a-6$, $a^{22}-4a^{20}-7a^{19}+3a^{18}+20a^{17}+18a^{16}-8a^{15}-29a^{14}-14a^{13}+5a^{12}-2a^{11}-20a^{10}-5a^{9}+30a^{8}+30a^{7}-3a^{6}-27a^{5}-15a^{4}+a^{3}+4a^{2}-a-2$, $36a^{22}-16a^{21}-144a^{20}-197a^{19}+216a^{18}+710a^{17}+400a^{16}-582a^{15}-1092a^{14}-239a^{13}+416a^{12}+52a^{11}-538a^{10}+155a^{9}+1261a^{8}+835a^{7}-487a^{6}-1128a^{5}-385a^{4}+244a^{3}+269a^{2}+28a-47$, $21a^{22}-7a^{21}-90a^{20}-121a^{19}+133a^{18}+448a^{17}+246a^{16}-397a^{15}-702a^{14}-134a^{13}+327a^{12}+61a^{11}-344a^{10}+94a^{9}+806a^{8}+521a^{7}-373a^{6}-751a^{5}-231a^{4}+197a^{3}+192a^{2}+19a-30$, $9a^{22}+a^{21}-44a^{20}-65a^{19}+52a^{18}+225a^{17}+150a^{16}-188a^{15}-357a^{14}-93a^{13}+174a^{12}+34a^{11}-192a^{10}+24a^{9}+402a^{8}+302a^{7}-179a^{6}-374a^{5}-122a^{4}+97a^{3}+94a^{2}+3a-16$, $35a^{22}-15a^{21}-147a^{20}-191a^{19}+236a^{18}+725a^{17}+360a^{16}-685a^{15}-1128a^{14}-153a^{13}+558a^{12}+91a^{11}-565a^{10}+190a^{9}+1315a^{8}+780a^{7}-678a^{6}-1227a^{5}-318a^{4}+358a^{3}+316a^{2}+18a-56$, $9a^{22}-4a^{21}-36a^{20}-49a^{19}+53a^{18}+178a^{17}+101a^{16}-144a^{15}-274a^{14}-65a^{13}+105a^{12}+14a^{11}-130a^{10}+38a^{9}+313a^{8}+212a^{7}-122a^{6}-282a^{5}-100a^{4}+62a^{3}+68a^{2}+8a-12$, $19a^{22}-9a^{21}-82a^{20}-97a^{19}+140a^{18}+398a^{17}+156a^{16}-415a^{15}-607a^{14}-9a^{13}+358a^{12}+30a^{11}-332a^{10}+116a^{9}+740a^{8}+358a^{7}-448a^{6}-676a^{5}-94a^{4}+258a^{3}+173a^{2}-14a-46$, $28a^{22}-8a^{21}-118a^{20}-168a^{19}+162a^{18}+590a^{17}+363a^{16}-473a^{15}-918a^{14}-231a^{13}+365a^{12}+58a^{11}-463a^{10}+100a^{9}+1046a^{8}+740a^{7}-402a^{6}-946a^{5}-336a^{4}+204a^{3}+226a^{2}+22a-38$, $2a^{22}-4a^{21}-8a^{20}+6a^{19}+31a^{18}+11a^{17}-54a^{16}-59a^{15}+33a^{14}+89a^{13}+5a^{12}-70a^{11}-14a^{10}+90a^{9}+23a^{8}-97a^{7}-80a^{6}+49a^{5}+84a^{4}-26a^{2}-20a+12$, $9a^{22}-5a^{21}-34a^{20}-48a^{19}+56a^{18}+169a^{17}+95a^{16}-144a^{15}-263a^{14}-54a^{13}+98a^{12}+18a^{11}-131a^{10}+45a^{9}+298a^{8}+200a^{7}-123a^{6}-274a^{5}-93a^{4}+61a^{3}+68a^{2}+8a-11$, $9a^{22}-5a^{21}-36a^{20}-46a^{19}+62a^{18}+176a^{17}+82a^{16}-167a^{15}-272a^{14}-31a^{13}+121a^{12}+18a^{11}-136a^{10}+60a^{9}+320a^{8}+180a^{7}-157a^{6}-295a^{5}-70a^{4}+73a^{3}+77a^{2}+a-9$, $a^{22}+7a^{21}-12a^{20}-32a^{19}-17a^{18}+86a^{17}+124a^{16}-16a^{15}-181a^{14}-150a^{13}+79a^{12}+82a^{11}-51a^{10}-97a^{9}+142a^{8}+239a^{7}+8a^{6}-196a^{5}-166a^{4}+49a^{3}+62a^{2}+33a-21$, $20a^{22}-8a^{21}-85a^{20}-112a^{19}+138a^{18}+421a^{17}+208a^{16}-406a^{15}-655a^{14}-73a^{13}+331a^{12}+49a^{11}-345a^{10}+118a^{9}+769a^{8}+445a^{7}-407a^{6}-711a^{5}-160a^{4}+211a^{3}+183a^{2}+a-31$
| 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: | \( 973481536934 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{7}\cdot 973481536934 \cdot 1}{2\cdot\sqrt{9554598217771454170076503812873306955636}}\cr\approx \mathstrut & 0.985645057136508 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*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 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*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 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*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 - x^22 - 4*x^21 - 3*x^20 + 10*x^19 + 17*x^18 - 2*x^17 - 26*x^16 - 21*x^15 + 15*x^14 + 19*x^13 - 7*x^12 - 18*x^11 + 15*x^10 + 35*x^9 - 33*x^7 - 24*x^6 + 12*x^5 + 16*x^4 + 3*x^3 - 5*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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.17.0.1 | $x^{17} + 2 x + 17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(432841815513398891\)
| $\Q_{432841815513398891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(290\!\cdots\!621\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |