Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*x^2 - 2*x + 1)
gp: K = bnfinit(y^23 - y^22 - y^21 - y^20 + y^19 + 8*y^18 - 3*y^17 - 9*y^16 - 6*y^15 + 5*y^14 + 23*y^13 - y^12 - 23*y^11 - 13*y^10 + 12*y^9 + 27*y^8 - 3*y^7 - 20*y^6 - 6*y^5 + 9*y^4 + 7*y^3 - 4*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - x^22 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - x^22 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*x^2 - 2*x + 1)
\( x^{23} - x^{22} - x^{21} - x^{20} + x^{19} + 8 x^{18} - 3 x^{17} - 9 x^{16} - 6 x^{15} + 5 x^{14} + \cdots + 1 \)
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: | $[3, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(96132210542339215127450630436634453\)
\(\medspace = 19\cdot 28201\cdot 55057\cdot 3258654194756619414566191\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.19\) | 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: | $19^{1/2}28201^{1/2}55057^{1/2}3258654194756619414566191^{1/2}\approx 3.1005194813504915e+17$ | ||
| Ramified primes: |
\(19\), \(28201\), \(55057\), \(3258654194756619414566191\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{96132\!\cdots\!34453}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $12$ | 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: |
$a$, $a^{22}-a^{21}-2a^{19}+a^{18}+6a^{17}-2a^{16}-3a^{15}-8a^{14}+2a^{13}+15a^{12}+a^{11}-8a^{10}-12a^{9}+4a^{8}+15a^{7}+a^{6}-5a^{5}-5a^{4}+4a^{3}+2a^{2}$, $a^{22}-a^{21}-a^{20}-a^{19}+a^{18}+8a^{17}-3a^{16}-9a^{15}-6a^{14}+5a^{13}+23a^{12}-a^{11}-23a^{10}-13a^{9}+12a^{8}+27a^{7}-3a^{6}-20a^{5}-6a^{4}+9a^{3}+7a^{2}-4a-1$, $3a^{22}-a^{21}-4a^{20}-5a^{19}+24a^{17}+6a^{16}-26a^{15}-32a^{14}-2a^{13}+68a^{12}+37a^{11}-54a^{10}-69a^{9}+3a^{8}+85a^{7}+37a^{6}-47a^{5}-42a^{4}+10a^{3}+29a^{2}+a-8$, $a^{22}-a^{21}-2a^{19}+a^{18}+6a^{17}-2a^{16}-3a^{15}-8a^{14}+2a^{13}+15a^{12}+a^{11}-8a^{10}-12a^{9}+4a^{8}+15a^{7}+a^{6}-5a^{5}-5a^{4}+4a^{3}+2a^{2}-a+1$, $8a^{22}-5a^{21}-9a^{20}-12a^{19}+3a^{18}+64a^{17}-66a^{15}-74a^{14}+8a^{13}+182a^{12}+60a^{11}-147a^{10}-158a^{9}+27a^{8}+219a^{7}+61a^{6}-123a^{5}-95a^{4}+30a^{3}+66a^{2}-3a-16$, $4a^{22}-3a^{21}-4a^{20}-6a^{19}+2a^{18}+32a^{17}-3a^{16}-31a^{15}-36a^{14}+5a^{13}+91a^{12}+24a^{11}-70a^{10}-76a^{9}+12a^{8}+107a^{7}+27a^{6}-56a^{5}-46a^{4}+10a^{3}+31a^{2}-6$, $12a^{22}-8a^{21}-11a^{20}-20a^{19}+4a^{18}+92a^{17}-a^{16}-82a^{15}-113a^{14}+2a^{13}+252a^{12}+87a^{11}-175a^{10}-223a^{9}+15a^{8}+285a^{7}+89a^{6}-134a^{5}-126a^{4}+26a^{3}+72a^{2}-4a-13$, $10a^{22}-11a^{21}-2a^{20}-16a^{19}+8a^{18}+68a^{17}-33a^{16}-37a^{15}-69a^{14}+22a^{13}+173a^{12}-20a^{11}-97a^{10}-106a^{9}+50a^{8}+169a^{7}-26a^{6}-60a^{5}-42a^{4}+39a^{3}+19a^{2}-21a+7$, $9a^{22}-7a^{21}-6a^{20}-15a^{19}+4a^{18}+65a^{17}-8a^{16}-51a^{15}-77a^{14}+6a^{13}+170a^{12}+43a^{11}-112a^{10}-140a^{9}+22a^{8}+179a^{7}+40a^{6}-78a^{5}-71a^{4}+27a^{3}+33a^{2}-8a-2$, $9a^{22}-4a^{21}-12a^{20}-14a^{19}+73a^{17}+13a^{16}-78a^{15}-89a^{14}-8a^{13}+201a^{12}+104a^{11}-158a^{10}-188a^{9}+a^{8}+230a^{7}+105a^{6}-124a^{5}-110a^{4}+17a^{3}+60a^{2}+4a-13$, $8a^{22}-22a^{21}+11a^{20}-7a^{19}+28a^{18}+45a^{17}-119a^{16}-a^{15}+9a^{14}+121a^{13}+113a^{12}-257a^{11}-96a^{10}+71a^{9}+218a^{8}+78a^{7}-272a^{6}-63a^{5}+74a^{4}+115a^{3}-29a^{2}-60a+23$
| 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: | \( 177401273.772 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 177401273.772 \cdot 1}{2\cdot\sqrt{96132210542339215127450630436634453}}\cr\approx \mathstrut & 0.219472921240 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*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 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*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 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*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 - x^21 - x^20 + x^19 + 8*x^18 - 3*x^17 - 9*x^16 - 6*x^15 + 5*x^14 + 23*x^13 - x^12 - 23*x^11 - 13*x^10 + 12*x^9 + 27*x^8 - 3*x^7 - 20*x^6 - 6*x^5 + 9*x^4 + 7*x^3 - 4*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 $S_{23}$ |
| Character table for $S_{23}$ |
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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | $21{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $23$ | $19{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.21.0.1 | $x^{21} + x^{9} + 12 x^{8} + 15 x^{7} + 12 x^{6} + x^{5} + 10 x^{4} + 7 x^{3} + 12 x^{2} + 16 x + 17$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
|
\(28201\)
| $\Q_{28201}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{28201}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{28201}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(55057\)
| $\Q_{55057}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(325\!\cdots\!191\)
| $\Q_{32\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{32\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |