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Properties

Label 23.13.291...844.1
Degree $23$
Signature $[13, 5]$
Discriminant $-2.915\times 10^{42}$
Root discriminant \(70.19\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{23}$ (as 23T7)

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Show commands: Magma / Oscar / PariGP / SageMath

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 + 19*x^3 + 39*x^2 - 3*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 + 19*y^3 + 39*y^2 - 3*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 + 19*x^3 + 39*x^2 - 3*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 + 19*x^3 + 39*x^2 - 3*x - 1)
 

\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} + \cdots - 1 \) Copy content Toggle raw display

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:  $[13, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2915266807970198336219632083292475251009844\) \(\medspace = -\,2^{2}\cdot 773\cdot 2479717146827\cdot 380221505923121334124604891\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(70.19\)
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:  not computed
Ramified primes:   \(2\), \(773\), \(2479717146827\), \(380221505923121334124604891\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{-72881\!\cdots\!52461}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$ Copy content Toggle raw display

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a$, $a^{2}-a-1$, $a^{22}-6a^{21}-4a^{20}+84a^{19}-66a^{18}-488a^{17}+660a^{16}+1522a^{15}-2645a^{14}-2750a^{13}+5813a^{12}+2878a^{11}-7515a^{10}-1595a^{9}+5690a^{8}+285a^{7}-2376a^{6}+127a^{5}+481a^{4}-56a^{3}-37a^{2}+2a$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8562a^{12}-2932a^{11}-10385a^{10}+5893a^{9}+7263a^{8}-5316a^{7}-2638a^{6}+2371a^{5}+349a^{4}-450a^{3}+15a^{2}+18a-1$, $a^{3}-a^{2}-3a$, $a^{22}-7a^{21}+5a^{20}+68a^{19}-145a^{18}-195a^{17}+789a^{16}-96a^{15}-1788a^{14}+1566a^{13}+1404a^{12}-3158a^{11}+1157a^{10}+2581a^{9}-2977a^{8}-637a^{7}+2124a^{6}-248a^{5}-636a^{4}+139a^{3}+72a^{2}-14a-2$, $a^{21}-9a^{20}+18a^{19}+66a^{18}-291a^{17}+17a^{16}+1405a^{15}-1468a^{14}-2937a^{13}+5494a^{12}+2111a^{11}-9044a^{10}+1562a^{9}+7340a^{8}-3337a^{7}-2719a^{6}+1740a^{5}+305a^{4}-276a^{3}+14a^{2}+12a-1$, $a^{22}-7a^{21}+a^{20}+94a^{19}-146a^{18}-506a^{17}+1215a^{16}+1345a^{15}-4833a^{14}-1565a^{13}+11189a^{12}-490a^{11}-15958a^{10}+3793a^{9}+13988a^{8}-4761a^{7}-7121a^{6}+2760a^{5}+1826a^{4}-706a^{3}-171a^{2}+40a+7$, $2a^{22}-16a^{21}+21a^{20}+148a^{19}-444a^{18}-326a^{17}+2479a^{16}-1024a^{15}-6443a^{14}+6498a^{13}+8225a^{12}-13400a^{11}-3926a^{10}+13632a^{9}-1607a^{8}-6980a^{7}+2465a^{6}+1521a^{5}-843a^{4}-57a^{3}+80a^{2}-3a-3$, $a^{21}-6a^{20}-2a^{19}+72a^{18}-69a^{17}-349a^{16}+514a^{15}+896a^{14}-1625a^{13}-1383a^{12}+2859a^{11}+1420a^{10}-3130a^{9}-940a^{8}+2236a^{7}+205a^{6}-948a^{5}+148a^{4}+164a^{3}-65a^{2}+3a+2$, $3a^{22}-25a^{21}+39a^{20}+214a^{19}-735a^{18}-309a^{17}+3884a^{16}-2492a^{15}-9381a^{14}+11997a^{13}+10340a^{12}-22494a^{11}-2343a^{10}+21160a^{9}-5093a^{8}-10018a^{7}+4504a^{6}+2051a^{5}-1335a^{4}-87a^{3}+128a^{2}-8a-3$, $4a^{22}-32a^{21}+39a^{20}+317a^{19}-900a^{18}-875a^{17}+5392a^{16}-1236a^{15}-15547a^{14}+12236a^{13}+23932a^{12}-29231a^{11}-18785a^{10}+34766a^{9}+4768a^{8}-22262a^{7}+2912a^{6}+7205a^{5}-2257a^{4}-877a^{3}+457a^{2}-19a-14$, $a^{22}-7a^{21}+3a^{20}+81a^{19}-146a^{18}-347a^{17}+1000a^{16}+588a^{15}-3242a^{14}+135a^{13}+5893a^{12}-1991a^{11}-6313a^{10}+3029a^{9}+3980a^{8}-2069a^{7}-1395a^{6}+651a^{5}+229a^{4}-68a^{3}-9a^{2}-4a-2$, $a^{22}-8a^{21}+7a^{20}+98a^{19}-233a^{18}-423a^{17}+1713a^{16}+479a^{15}-6195a^{14}+2069a^{13}+12643a^{12}-8676a^{11}-14805a^{10}+14228a^{9}+9224a^{8}-12051a^{7}-2147a^{6}+5210a^{5}-431a^{4}-957a^{3}+229a^{2}+31a-11$, $a^{20}-7a^{19}+4a^{18}+75a^{17}-148a^{16}-274a^{15}+925a^{14}+239a^{13}-2666a^{12}+950a^{11}+4042a^{10}-2892a^{9}-3173a^{8}+3281a^{7}+1062a^{6}-1732a^{5}+12a^{4}+380a^{3}-61a^{2}-19a+4$, $5a^{22}-33a^{21}-4a^{20}+444a^{19}-575a^{18}-2404a^{17}+4898a^{16}+6532a^{15}-19032a^{14}-8501a^{13}+41979a^{12}+1617a^{11}-55737a^{10}+9662a^{9}+44510a^{8}-12407a^{7}-20376a^{6}+6344a^{5}+4783a^{4}-1316a^{3}-473a^{2}+56a+13$, $3a^{21}-24a^{20}+32a^{19}+218a^{18}-660a^{17}-456a^{16}+3604a^{15}-1566a^{14}-9085a^{13}+9250a^{12}+11098a^{11}-18011a^{10}-5003a^{9}+16948a^{8}-1704a^{7}-7738a^{6}+2259a^{5}+1403a^{4}-544a^{3}-35a^{2}+8a$ Copy content Toggle raw display (assuming GRH)
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:  \( 21282525078200 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{5}\cdot 21282525078200 \cdot 1}{2\cdot\sqrt{2915266807970198336219632083292475251009844}}\cr\approx \mathstrut & 0.499969244463958 \end{aligned}\] (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 + 19*x^3 + 39*x^2 - 3*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 + 19*x^3 + 39*x^2 - 3*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 + 19*x^3 + 39*x^2 - 3*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 + 19*x^3 + 39*x^2 - 3*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

$S_{23}$ (as 23T7):

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 $16{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $18{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ $16{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ $15{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $16{,}\,{\href{/padicField/41.7.0.1}{7} }$ $19{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ $23$ ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.21.0.1$x^{21} + x^{6} + x^{5} + x^{2} + 1$$1$$21$$0$$C_{21}$$[\ ]^{21}$
\(773\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $17$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(2479717146827\) Copy content Toggle raw display $\Q_{2479717146827}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(380\!\cdots\!891\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $19$$1$$19$$0$$C_{19}$$[\ ]^{19}$