Properties

Label 23.3.131...376.1
Degree $23$
Signature $[3, 10]$
Discriminant $1.317\times 10^{75}$
Root discriminant \(1845.32\)
Ramified primes $2,3,11,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_{23}$ (as 23T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 89424*x - 85536)
 
gp: K = bnfinit(y^23 - 89424*y - 85536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 89424*x - 85536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 89424*x - 85536)
 

\( x^{23} - 89424x - 85536 \) 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:  $[3, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1317269425233583227283427390293559977631759906982803446016843999202169061376\) \(\medspace = 2^{30}\cdot 3^{22}\cdot 11^{22}\cdot 23^{24}\) 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:  \(1845.32\)
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:   \(2\), \(3\), \(11\), \(23\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{6}a^{7}$, $\frac{1}{6}a^{8}$, $\frac{1}{6}a^{9}$, $\frac{1}{18}a^{10}$, $\frac{1}{36}a^{11}$, $\frac{1}{36}a^{12}$, $\frac{1}{36}a^{13}$, $\frac{1}{108}a^{14}$, $\frac{1}{216}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{216}a^{16}-\frac{1}{6}a^{5}$, $\frac{1}{216}a^{17}$, $\frac{1}{216}a^{18}$, $\frac{1}{648}a^{19}$, $\frac{1}{1296}a^{20}-\frac{1}{12}a^{9}$, $\frac{1}{1296}a^{21}-\frac{1}{36}a^{10}$, $\frac{1}{16848}a^{22}-\frac{1}{16848}a^{21}+\frac{1}{16848}a^{20}+\frac{1}{1404}a^{19}-\frac{1}{1404}a^{18}+\frac{1}{1404}a^{17}-\frac{1}{1404}a^{16}+\frac{1}{1404}a^{15}-\frac{1}{1404}a^{14}-\frac{1}{117}a^{13}+\frac{1}{117}a^{12}-\frac{1}{117}a^{11}-\frac{1}{52}a^{10}+\frac{1}{52}a^{9}+\frac{5}{78}a^{8}-\frac{5}{78}a^{7}+\frac{5}{78}a^{6}+\frac{4}{39}a^{5}+\frac{3}{13}a^{4}-\frac{3}{13}a^{3}+\frac{3}{13}a^{2}-\frac{3}{13}a-\frac{1}{13}$ Copy content Toggle raw display

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:  $12$
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:   $\frac{17\!\cdots\!13}{702}a^{22}-\frac{61\!\cdots\!19}{16848}a^{21}+\frac{81\!\cdots\!99}{16848}a^{20}-\frac{12\!\cdots\!71}{234}a^{19}+\frac{15\!\cdots\!11}{39}a^{18}+\frac{67\!\cdots\!83}{312}a^{17}-\frac{39\!\cdots\!17}{234}a^{16}+\frac{32\!\cdots\!47}{702}a^{15}-\frac{14\!\cdots\!67}{1404}a^{14}+\frac{14\!\cdots\!09}{78}a^{13}-\frac{36\!\cdots\!79}{117}a^{12}+\frac{11\!\cdots\!83}{234}a^{11}-\frac{31\!\cdots\!23}{468}a^{10}+\frac{12\!\cdots\!09}{156}a^{9}-\frac{56\!\cdots\!57}{78}a^{8}+\frac{96\!\cdots\!73}{78}a^{7}+\frac{39\!\cdots\!47}{26}a^{6}-\frac{19\!\cdots\!91}{39}a^{5}+\frac{15\!\cdots\!74}{13}a^{4}-\frac{29\!\cdots\!32}{13}a^{3}+\frac{51\!\cdots\!89}{13}a^{2}-\frac{82\!\cdots\!26}{13}a-\frac{16\!\cdots\!95}{13}$, $\frac{43\!\cdots\!47}{16848}a^{22}+\frac{31\!\cdots\!11}{5616}a^{21}+\frac{24\!\cdots\!93}{5616}a^{20}+\frac{16\!\cdots\!51}{4212}a^{19}+\frac{22\!\cdots\!03}{2808}a^{18}+\frac{27\!\cdots\!72}{351}a^{17}+\frac{30\!\cdots\!21}{1404}a^{16}+\frac{12\!\cdots\!33}{2808}a^{15}+\frac{10\!\cdots\!14}{351}a^{14}+\frac{24\!\cdots\!85}{468}a^{13}+\frac{51\!\cdots\!81}{468}a^{12}-\frac{62\!\cdots\!83}{156}a^{11}-\frac{33\!\cdots\!73}{156}a^{10}-\frac{10\!\cdots\!09}{52}a^{9}-\frac{47\!\cdots\!88}{39}a^{8}-\frac{22\!\cdots\!01}{78}a^{7}-\frac{13\!\cdots\!81}{39}a^{6}-\frac{28\!\cdots\!22}{39}a^{5}-\frac{41\!\cdots\!55}{26}a^{4}-\frac{19\!\cdots\!70}{13}a^{3}-\frac{24\!\cdots\!52}{13}a^{2}-\frac{52\!\cdots\!90}{13}a-\frac{33\!\cdots\!53}{13}$, $\frac{24\!\cdots\!27}{16848}a^{22}-\frac{39\!\cdots\!43}{16848}a^{21}+\frac{20\!\cdots\!31}{5616}a^{20}-\frac{63\!\cdots\!21}{1053}a^{19}+\frac{27\!\cdots\!15}{2808}a^{18}-\frac{43\!\cdots\!43}{2808}a^{17}+\frac{11\!\cdots\!83}{468}a^{16}-\frac{12\!\cdots\!27}{312}a^{15}+\frac{91\!\cdots\!05}{1404}a^{14}-\frac{16\!\cdots\!49}{156}a^{13}+\frac{39\!\cdots\!05}{234}a^{12}-\frac{12\!\cdots\!95}{468}a^{11}+\frac{22\!\cdots\!21}{52}a^{10}-\frac{11\!\cdots\!33}{156}a^{9}+\frac{44\!\cdots\!12}{39}a^{8}-\frac{71\!\cdots\!73}{39}a^{7}+\frac{23\!\cdots\!19}{78}a^{6}-\frac{62\!\cdots\!16}{13}a^{5}+\frac{20\!\cdots\!65}{26}a^{4}-\frac{16\!\cdots\!67}{13}a^{3}+\frac{26\!\cdots\!37}{13}a^{2}-\frac{41\!\cdots\!20}{13}a-\frac{98\!\cdots\!11}{13}$, $\frac{20\!\cdots\!27}{16848}a^{22}+\frac{88\!\cdots\!87}{4212}a^{21}+\frac{60\!\cdots\!71}{16848}a^{20}+\frac{26\!\cdots\!93}{4212}a^{19}+\frac{29\!\cdots\!77}{2808}a^{18}+\frac{51\!\cdots\!71}{2808}a^{17}+\frac{12\!\cdots\!57}{39}a^{16}+\frac{83\!\cdots\!57}{156}a^{15}+\frac{32\!\cdots\!42}{351}a^{14}+\frac{24\!\cdots\!49}{156}a^{13}+\frac{10\!\cdots\!77}{39}a^{12}+\frac{21\!\cdots\!61}{468}a^{11}+\frac{92\!\cdots\!93}{117}a^{10}+\frac{21\!\cdots\!83}{156}a^{9}+\frac{30\!\cdots\!01}{13}a^{8}+\frac{30\!\cdots\!73}{78}a^{7}+\frac{88\!\cdots\!49}{13}a^{6}+\frac{15\!\cdots\!12}{13}a^{5}+\frac{25\!\cdots\!71}{13}a^{4}+\frac{44\!\cdots\!13}{13}a^{3}+\frac{76\!\cdots\!51}{13}a^{2}+\frac{13\!\cdots\!11}{13}a+\frac{79\!\cdots\!33}{13}$, $\frac{23\!\cdots\!95}{8424}a^{22}+\frac{10\!\cdots\!13}{936}a^{21}+\frac{38\!\cdots\!09}{16848}a^{20}+\frac{73\!\cdots\!17}{2106}a^{19}+\frac{32\!\cdots\!21}{936}a^{18}-\frac{89\!\cdots\!17}{2808}a^{17}-\frac{30\!\cdots\!01}{2808}a^{16}-\frac{86\!\cdots\!01}{2808}a^{15}-\frac{40\!\cdots\!43}{702}a^{14}-\frac{88\!\cdots\!64}{117}a^{13}-\frac{59\!\cdots\!73}{117}a^{12}+\frac{19\!\cdots\!25}{26}a^{11}+\frac{84\!\cdots\!17}{234}a^{10}+\frac{12\!\cdots\!75}{156}a^{9}+\frac{17\!\cdots\!96}{13}a^{8}+\frac{57\!\cdots\!92}{39}a^{7}+\frac{23\!\cdots\!55}{78}a^{6}-\frac{26\!\cdots\!17}{78}a^{5}-\frac{27\!\cdots\!29}{26}a^{4}-\frac{27\!\cdots\!44}{13}a^{3}-\frac{38\!\cdots\!44}{13}a^{2}-\frac{31\!\cdots\!43}{13}a-\frac{10\!\cdots\!81}{13}$, $\frac{20\!\cdots\!07}{8424}a^{22}-\frac{12\!\cdots\!27}{312}a^{21}+\frac{29\!\cdots\!75}{468}a^{20}-\frac{84\!\cdots\!65}{8424}a^{19}+\frac{22\!\cdots\!27}{1404}a^{18}-\frac{73\!\cdots\!29}{2808}a^{17}+\frac{59\!\cdots\!23}{1404}a^{16}-\frac{95\!\cdots\!65}{1404}a^{15}+\frac{38\!\cdots\!51}{351}a^{14}-\frac{82\!\cdots\!19}{468}a^{13}+\frac{66\!\cdots\!85}{234}a^{12}-\frac{53\!\cdots\!30}{117}a^{11}+\frac{86\!\cdots\!48}{117}a^{10}-\frac{15\!\cdots\!26}{13}a^{9}+\frac{74\!\cdots\!65}{39}a^{8}-\frac{12\!\cdots\!76}{39}a^{7}+\frac{12\!\cdots\!71}{26}a^{6}-\frac{31\!\cdots\!06}{39}a^{5}+\frac{16\!\cdots\!43}{13}a^{4}-\frac{27\!\cdots\!51}{13}a^{3}+\frac{43\!\cdots\!71}{13}a^{2}-\frac{70\!\cdots\!26}{13}a-\frac{16\!\cdots\!61}{13}$, $\frac{13\!\cdots\!97}{2106}a^{22}-\frac{16\!\cdots\!11}{4212}a^{21}-\frac{15\!\cdots\!95}{16848}a^{20}+\frac{25\!\cdots\!59}{8424}a^{19}-\frac{13\!\cdots\!71}{351}a^{18}-\frac{11\!\cdots\!34}{351}a^{17}+\frac{78\!\cdots\!59}{702}a^{16}-\frac{63\!\cdots\!73}{2808}a^{15}+\frac{58\!\cdots\!82}{351}a^{14}+\frac{12\!\cdots\!17}{468}a^{13}-\frac{48\!\cdots\!41}{468}a^{12}+\frac{55\!\cdots\!41}{39}a^{11}-\frac{17\!\cdots\!54}{117}a^{10}-\frac{56\!\cdots\!43}{156}a^{9}+\frac{31\!\cdots\!73}{39}a^{8}-\frac{89\!\cdots\!10}{13}a^{7}-\frac{58\!\cdots\!29}{78}a^{6}+\frac{45\!\cdots\!44}{13}a^{5}-\frac{13\!\cdots\!89}{26}a^{4}+\frac{18\!\cdots\!55}{13}a^{3}+\frac{14\!\cdots\!38}{13}a^{2}-\frac{36\!\cdots\!10}{13}a-\frac{38\!\cdots\!41}{13}$, $\frac{11\!\cdots\!11}{624}a^{22}-\frac{39\!\cdots\!89}{5616}a^{21}+\frac{92\!\cdots\!29}{5616}a^{20}-\frac{77\!\cdots\!49}{234}a^{19}+\frac{16\!\cdots\!33}{2808}a^{18}-\frac{26\!\cdots\!03}{2808}a^{17}+\frac{38\!\cdots\!63}{2808}a^{16}-\frac{12\!\cdots\!03}{702}a^{15}+\frac{27\!\cdots\!31}{1404}a^{14}-\frac{97\!\cdots\!51}{78}a^{13}-\frac{14\!\cdots\!76}{117}a^{12}+\frac{11\!\cdots\!77}{156}a^{11}-\frac{97\!\cdots\!45}{52}a^{10}+\frac{20\!\cdots\!09}{52}a^{9}-\frac{94\!\cdots\!23}{13}a^{8}+\frac{94\!\cdots\!13}{78}a^{7}-\frac{14\!\cdots\!29}{78}a^{6}+\frac{65\!\cdots\!79}{26}a^{5}-\frac{38\!\cdots\!98}{13}a^{4}+\frac{32\!\cdots\!73}{13}a^{3}+\frac{16\!\cdots\!07}{13}a^{2}-\frac{86\!\cdots\!36}{13}a+\frac{46\!\cdots\!89}{13}$, $\frac{22\!\cdots\!33}{16848}a^{22}+\frac{48\!\cdots\!11}{1872}a^{21}+\frac{16\!\cdots\!29}{52}a^{20}+\frac{12\!\cdots\!29}{4212}a^{19}-\frac{20\!\cdots\!47}{702}a^{18}-\frac{40\!\cdots\!43}{2808}a^{17}-\frac{24\!\cdots\!21}{702}a^{16}-\frac{17\!\cdots\!59}{2808}a^{15}-\frac{45\!\cdots\!51}{702}a^{14}-\frac{49\!\cdots\!91}{156}a^{13}+\frac{31\!\cdots\!13}{234}a^{12}+\frac{21\!\cdots\!91}{468}a^{11}+\frac{42\!\cdots\!13}{468}a^{10}+\frac{11\!\cdots\!97}{78}a^{9}+\frac{29\!\cdots\!69}{26}a^{8}-\frac{45\!\cdots\!55}{13}a^{7}-\frac{18\!\cdots\!21}{39}a^{6}-\frac{50\!\cdots\!29}{39}a^{5}-\frac{57\!\cdots\!99}{26}a^{4}-\frac{38\!\cdots\!56}{13}a^{3}-\frac{19\!\cdots\!31}{13}a^{2}+\frac{47\!\cdots\!49}{13}a+\frac{44\!\cdots\!07}{13}$, $\frac{44\!\cdots\!51}{5616}a^{22}-\frac{70\!\cdots\!01}{16848}a^{21}-\frac{11\!\cdots\!57}{16848}a^{20}-\frac{55\!\cdots\!09}{1404}a^{19}-\frac{38\!\cdots\!25}{234}a^{18}-\frac{15\!\cdots\!75}{1404}a^{17}+\frac{30\!\cdots\!31}{2808}a^{16}+\frac{86\!\cdots\!09}{351}a^{15}+\frac{13\!\cdots\!34}{117}a^{14}+\frac{16\!\cdots\!10}{117}a^{13}-\frac{16\!\cdots\!38}{117}a^{12}-\frac{49\!\cdots\!51}{468}a^{11}-\frac{11\!\cdots\!43}{52}a^{10}+\frac{70\!\cdots\!19}{156}a^{9}+\frac{27\!\cdots\!59}{39}a^{8}+\frac{12\!\cdots\!83}{78}a^{7}+\frac{70\!\cdots\!63}{78}a^{6}-\frac{26\!\cdots\!67}{78}a^{5}-\frac{94\!\cdots\!77}{13}a^{4}-\frac{14\!\cdots\!09}{13}a^{3}+\frac{10\!\cdots\!07}{13}a^{2}+\frac{32\!\cdots\!33}{13}a+\frac{14\!\cdots\!47}{13}$, $\frac{73\!\cdots\!15}{2808}a^{22}-\frac{39\!\cdots\!65}{5616}a^{21}+\frac{11\!\cdots\!73}{936}a^{20}-\frac{11\!\cdots\!85}{8424}a^{19}+\frac{14\!\cdots\!55}{2808}a^{18}+\frac{67\!\cdots\!09}{2808}a^{17}-\frac{36\!\cdots\!75}{468}a^{16}+\frac{17\!\cdots\!76}{117}a^{15}-\frac{27\!\cdots\!83}{1404}a^{14}+\frac{57\!\cdots\!69}{468}a^{13}+\frac{48\!\cdots\!55}{26}a^{12}-\frac{65\!\cdots\!31}{78}a^{11}+\frac{83\!\cdots\!77}{468}a^{10}-\frac{99\!\cdots\!22}{39}a^{9}+\frac{56\!\cdots\!91}{26}a^{8}+\frac{26\!\cdots\!61}{26}a^{7}-\frac{67\!\cdots\!33}{78}a^{6}+\frac{80\!\cdots\!50}{39}a^{5}-\frac{42\!\cdots\!52}{13}a^{4}+\frac{43\!\cdots\!83}{13}a^{3}-\frac{31\!\cdots\!06}{13}a^{2}-\frac{10\!\cdots\!88}{13}a-\frac{23\!\cdots\!01}{13}$, $\frac{35\!\cdots\!69}{2808}a^{22}-\frac{57\!\cdots\!69}{1053}a^{21}+\frac{14\!\cdots\!37}{1872}a^{20}-\frac{47\!\cdots\!05}{2808}a^{19}-\frac{56\!\cdots\!09}{312}a^{18}+\frac{60\!\cdots\!29}{1404}a^{17}-\frac{11\!\cdots\!43}{2808}a^{16}-\frac{95\!\cdots\!81}{2808}a^{15}+\frac{85\!\cdots\!95}{468}a^{14}-\frac{33\!\cdots\!51}{117}a^{13}+\frac{25\!\cdots\!05}{234}a^{12}+\frac{66\!\cdots\!78}{117}a^{11}-\frac{19\!\cdots\!53}{13}a^{10}+\frac{24\!\cdots\!59}{156}a^{9}+\frac{63\!\cdots\!89}{78}a^{8}-\frac{47\!\cdots\!45}{78}a^{7}+\frac{27\!\cdots\!51}{26}a^{6}-\frac{41\!\cdots\!87}{78}a^{5}-\frac{45\!\cdots\!13}{26}a^{4}+\frac{67\!\cdots\!24}{13}a^{3}-\frac{78\!\cdots\!77}{13}a^{2}-\frac{20\!\cdots\!36}{13}a+\frac{11\!\cdots\!71}{13}$ 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:  \( 199639980384000000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 199639980384000000000000000000 \cdot 1}{2\cdot\sqrt{1317269425233583227283427390293559977631759906982803446016843999202169061376}}\cr\approx \mathstrut & 2.10993408300260 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 89424*x - 85536)
 
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 - 89424*x - 85536, 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 - 89424*x - 85536);
 
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 - 89424*x - 85536);
 
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

$A_{23}$ (as 23T6):

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 12926008369442488320000
The 641 conjugacy class representatives for $A_{23}$ are not computed
Character table for $A_{23}$ 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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $19{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R $20{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ $16{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ R $15{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $23$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ $20{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
Deg $22$$22$$1$$30$
\(3\) Copy content Toggle raw display 3.23.22.1$x^{23} + 3$$23$$1$$22$$C_{23}:C_{11}$$[\ ]_{23}^{11}$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.11.11.7$x^{11} + 44 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
11.11.11.9$x^{11} + 99 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
\(23\) Copy content Toggle raw display 23.23.24.4$x^{23} + 92 x^{2} + 23$$23$$1$$24$$C_{23}:C_{11}$$[12/11]_{11}$