Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 8*x - 4)
gp: K = bnfinit(y^23 + 8*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 + 8*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 8*x - 4)
\( x^{23} + 8x - 4 \)
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: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-28585240626865948858207093238982844088320\)
\(\medspace = -\,2^{22}\cdot 5\cdot 24986142523\cdot 54552257894632015274417\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.41\) | 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: | $2^{22/23}5^{1/2}24986142523^{1/2}54552257894632015274417^{1/2}\approx 1.6020745037248333e+17$ | ||
| Ramified primes: |
\(2\), \(5\), \(24986142523\), \(54552257894632015274417\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-68152\!\cdots\!70455}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{82}a^{22}-\frac{3}{41}a^{21}-\frac{5}{82}a^{20}-\frac{11}{82}a^{19}-\frac{8}{41}a^{18}+\frac{7}{41}a^{17}-\frac{1}{41}a^{16}+\frac{6}{41}a^{15}+\frac{5}{41}a^{14}-\frac{19}{82}a^{13}-\frac{9}{82}a^{12}-\frac{14}{41}a^{11}+\frac{2}{41}a^{10}-\frac{12}{41}a^{9}-\frac{10}{41}a^{8}+\frac{19}{41}a^{7}+\frac{9}{41}a^{6}-\frac{13}{41}a^{5}-\frac{4}{41}a^{4}-\frac{17}{41}a^{3}+\frac{20}{41}a^{2}+\frac{3}{41}a-\frac{14}{41}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $11$ | 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: |
$\frac{1}{41}a^{22}-\frac{6}{41}a^{21}-\frac{5}{41}a^{20}+\frac{19}{82}a^{19}+\frac{9}{82}a^{18}-\frac{13}{82}a^{17}-\frac{2}{41}a^{16}-\frac{17}{82}a^{15}+\frac{10}{41}a^{14}+\frac{3}{82}a^{13}+\frac{23}{82}a^{12}-\frac{28}{41}a^{11}+\frac{4}{41}a^{10}+\frac{17}{41}a^{9}+\frac{21}{41}a^{8}-\frac{44}{41}a^{7}+\frac{18}{41}a^{6}-\frac{26}{41}a^{5}+\frac{33}{41}a^{4}+\frac{7}{41}a^{3}-\frac{1}{41}a^{2}-\frac{35}{41}a+\frac{13}{41}$, $\frac{32}{41}a^{22}+\frac{13}{41}a^{21}+\frac{4}{41}a^{20}-\frac{7}{82}a^{19}+\frac{1}{82}a^{18}-\frac{3}{41}a^{17}-\frac{5}{82}a^{16}-\frac{11}{82}a^{15}-\frac{8}{41}a^{14}-\frac{27}{82}a^{13}-\frac{1}{41}a^{12}+\frac{6}{41}a^{11}+\frac{5}{41}a^{10}+\frac{11}{41}a^{9}+\frac{16}{41}a^{8}+\frac{27}{41}a^{7}+\frac{2}{41}a^{6}+\frac{29}{41}a^{5}-\frac{10}{41}a^{4}-\frac{22}{41}a^{3}-\frac{73}{41}a^{2}-\frac{13}{41}a+\frac{211}{41}$, $\frac{8}{41}a^{22}+\frac{27}{82}a^{21}+\frac{1}{41}a^{20}-\frac{6}{41}a^{19}-\frac{5}{41}a^{18}+\frac{19}{82}a^{17}+\frac{9}{82}a^{16}-\frac{13}{82}a^{15}+\frac{37}{82}a^{14}+\frac{12}{41}a^{13}-\frac{21}{82}a^{12}-\frac{19}{41}a^{11}-\frac{9}{41}a^{10}+\frac{13}{41}a^{9}-\frac{37}{41}a^{8}-\frac{65}{41}a^{7}-\frac{20}{41}a^{6}-\frac{44}{41}a^{5}-\frac{64}{41}a^{4}-\frac{67}{41}a^{3}-\frac{8}{41}a^{2}+\frac{48}{41}a-\frac{19}{41}$, $\frac{400}{41}a^{22}+\frac{407}{82}a^{21}+\frac{91}{41}a^{20}+\frac{28}{41}a^{19}+\frac{33}{82}a^{18}+\frac{24}{41}a^{17}+\frac{20}{41}a^{16}+\frac{3}{41}a^{15}+\frac{5}{82}a^{14}-\frac{15}{41}a^{13}-\frac{25}{82}a^{12}-\frac{7}{41}a^{11}+\frac{42}{41}a^{10}+\frac{35}{41}a^{9}-\frac{5}{41}a^{8}-\frac{52}{41}a^{7}-\frac{16}{41}a^{6}+\frac{14}{41}a^{5}+\frac{39}{41}a^{4}+\frac{12}{41}a^{3}+\frac{10}{41}a^{2}-\frac{19}{41}a+\frac{3109}{41}$, $\frac{156}{41}a^{22}+\frac{301}{82}a^{21}+\frac{122}{41}a^{20}+\frac{47}{41}a^{19}-\frac{36}{41}a^{18}-\frac{265}{82}a^{17}-\frac{419}{82}a^{16}-\frac{260}{41}a^{15}-\frac{529}{82}a^{14}-\frac{393}{82}a^{13}-\frac{92}{41}a^{12}+\frac{60}{41}a^{11}+\frac{214}{41}a^{10}+\frac{356}{41}a^{9}+\frac{447}{41}a^{8}+\frac{434}{41}a^{7}+\frac{348}{41}a^{6}+\frac{167}{41}a^{5}-\frac{59}{41}a^{4}-\frac{343}{41}a^{3}-\frac{566}{41}a^{2}-\frac{704}{41}a+\frac{511}{41}$, $\frac{1457}{82}a^{22}+\frac{385}{41}a^{21}+\frac{191}{41}a^{20}+\frac{84}{41}a^{19}+\frac{17}{82}a^{18}-\frac{10}{41}a^{17}-\frac{22}{41}a^{16}+\frac{9}{41}a^{15}+\frac{28}{41}a^{14}+\frac{115}{82}a^{13}+\frac{89}{82}a^{12}+\frac{20}{41}a^{11}-\frac{38}{41}a^{10}-\frac{59}{41}a^{9}-\frac{56}{41}a^{8}-\frac{33}{41}a^{7}+\frac{34}{41}a^{6}+\frac{83}{41}a^{5}+\frac{117}{41}a^{4}+\frac{36}{41}a^{3}-\frac{11}{41}a^{2}-\frac{139}{41}a+\frac{5719}{41}$, $\frac{5}{82}a^{22}+\frac{11}{82}a^{21}+\frac{8}{41}a^{20}+\frac{27}{82}a^{19}+\frac{1}{41}a^{18}-\frac{6}{41}a^{17}-\frac{5}{41}a^{16}-\frac{11}{41}a^{15}+\frac{9}{82}a^{14}-\frac{27}{41}a^{13}-\frac{2}{41}a^{12}-\frac{29}{41}a^{11}+\frac{10}{41}a^{10}+\frac{22}{41}a^{9}-\frac{9}{41}a^{8}+\frac{54}{41}a^{7}-\frac{37}{41}a^{6}+\frac{99}{41}a^{5}-\frac{61}{41}a^{4}+\frac{79}{41}a^{3}-\frac{105}{41}a^{2}+\frac{56}{41}a-\frac{29}{41}$, $\frac{7}{41}a^{22}+\frac{39}{82}a^{21}-\frac{29}{82}a^{20}-\frac{31}{82}a^{19}+\frac{11}{41}a^{18}+\frac{73}{82}a^{17}-\frac{14}{41}a^{16}-\frac{39}{41}a^{15}+\frac{29}{41}a^{14}+\frac{103}{82}a^{13}-\frac{22}{41}a^{12}-\frac{73}{41}a^{11}+\frac{69}{41}a^{10}+\frac{78}{41}a^{9}-\frac{58}{41}a^{8}-\frac{62}{41}a^{7}+\frac{85}{41}a^{6}+\frac{105}{41}a^{5}-\frac{138}{41}a^{4}-\frac{74}{41}a^{3}+\frac{116}{41}a^{2}+\frac{124}{41}a-\frac{73}{41}$, $\frac{24}{41}a^{22}+\frac{20}{41}a^{21}+\frac{47}{82}a^{20}+\frac{23}{41}a^{19}+\frac{26}{41}a^{18}+\frac{49}{41}a^{17}+\frac{109}{82}a^{16}+\frac{42}{41}a^{15}+\frac{111}{82}a^{14}+\frac{113}{82}a^{13}+\frac{101}{82}a^{12}+\frac{66}{41}a^{11}+\frac{55}{41}a^{10}+\frac{39}{41}a^{9}+\frac{53}{41}a^{8}+\frac{10}{41}a^{7}-\frac{19}{41}a^{6}+\frac{32}{41}a^{5}+\frac{13}{41}a^{4}-\frac{37}{41}a^{3}-\frac{24}{41}a^{2}-\frac{61}{41}a+\frac{107}{41}$, $\frac{39}{82}a^{22}+\frac{6}{41}a^{21}-\frac{31}{82}a^{20}-\frac{19}{82}a^{19}-\frac{9}{82}a^{18}-\frac{14}{41}a^{17}+\frac{45}{82}a^{16}+\frac{29}{41}a^{15}-\frac{10}{41}a^{14}-\frac{3}{82}a^{13}+\frac{59}{82}a^{12}-\frac{13}{41}a^{11}-\frac{86}{41}a^{10}-\frac{17}{41}a^{9}+\frac{20}{41}a^{8}+\frac{3}{41}a^{7}+\frac{64}{41}a^{6}+\frac{67}{41}a^{5}-\frac{33}{41}a^{4}-\frac{48}{41}a^{3}+\frac{83}{41}a^{2}-\frac{47}{41}a-\frac{13}{41}$, $\frac{91}{41}a^{22}-\frac{313}{82}a^{21}-\frac{86}{41}a^{20}+\frac{417}{82}a^{19}+\frac{102}{41}a^{18}-\frac{527}{82}a^{17}-\frac{241}{82}a^{16}+\frac{313}{41}a^{15}+\frac{221}{82}a^{14}-\frac{793}{82}a^{13}-\frac{285}{82}a^{12}+\frac{486}{41}a^{11}+\frac{159}{41}a^{10}-\frac{585}{41}a^{9}-\frac{139}{41}a^{8}+\frac{752}{41}a^{7}+\frac{162}{41}a^{6}-\frac{890}{41}a^{5}-\frac{154}{41}a^{4}+\frac{1088}{41}a^{3}+\frac{114}{41}a^{2}-\frac{1381}{41}a+\frac{609}{41}$
| 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: | \( 281108277991 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 281108277991 \cdot 1}{2\cdot\sqrt{28585240626865948858207093238982844088320}}\cr\approx \mathstrut & 1.00180053782095 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 + 8*x - 4)
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 + 8*x - 4, 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 + 8*x - 4);
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 + 8*x - 4);
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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | $19{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.23.22.1 | $x^{23} + 2$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(24986142523\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(545\!\cdots\!417\)
| $\Q_{54\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |