Properties

Label 23.3.823...336.1
Degree $23$
Signature $[3, 10]$
Discriminant $8.233\times 10^{73}$
Root discriminant \(1635.76\)
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 - 13248*x - 12672)
 
gp: K = bnfinit(y^23 - 13248*y - 12672, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 13248*x - 12672);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 13248*x - 12672)
 

\( x^{23} - 13248x - 12672 \) 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:   \(82329339077098951705214211893347498601984994186425215376052749950135566336\) \(\medspace = 2^{26}\cdot 3^{22}\cdot 11^{22}\cdot 23^{24}\) 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:  \(1635.76\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{8}a^{11}$, $\frac{1}{24}a^{12}$, $\frac{1}{48}a^{13}-\frac{1}{4}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{48}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{48}a^{15}$, $\frac{1}{48}a^{16}$, $\frac{1}{96}a^{17}-\frac{1}{4}a^{6}$, $\frac{1}{96}a^{18}-\frac{1}{4}a^{7}$, $\frac{1}{96}a^{19}$, $\frac{1}{96}a^{20}$, $\frac{1}{192}a^{21}-\frac{1}{8}a^{10}$, $\frac{1}{960}a^{22}-\frac{1}{960}a^{21}-\frac{1}{240}a^{20}+\frac{1}{240}a^{19}-\frac{1}{240}a^{18}+\frac{1}{240}a^{17}-\frac{1}{240}a^{16}+\frac{1}{240}a^{15}-\frac{1}{240}a^{14}+\frac{1}{240}a^{13}+\frac{1}{60}a^{12}+\frac{1}{40}a^{11}-\frac{1}{40}a^{10}-\frac{1}{10}a^{9}+\frac{1}{10}a^{8}+\frac{3}{20}a^{7}+\frac{1}{10}a^{6}-\frac{1}{10}a^{5}+\frac{1}{10}a^{4}-\frac{1}{10}a^{3}+\frac{1}{10}a^{2}+\frac{2}{5}a-\frac{1}{5}$ 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:   $\frac{57\!\cdots\!53}{320}a^{22}-\frac{22\!\cdots\!17}{480}a^{21}-\frac{68\!\cdots\!73}{480}a^{20}-\frac{87\!\cdots\!47}{480}a^{19}-\frac{13\!\cdots\!63}{480}a^{18}+\frac{31\!\cdots\!13}{80}a^{17}+\frac{13\!\cdots\!16}{15}a^{16}+\frac{44\!\cdots\!03}{5}a^{15}-\frac{77\!\cdots\!59}{240}a^{14}-\frac{68\!\cdots\!61}{240}a^{13}-\frac{61\!\cdots\!17}{120}a^{12}-\frac{14\!\cdots\!71}{40}a^{11}+\frac{24\!\cdots\!47}{5}a^{10}+\frac{38\!\cdots\!67}{20}a^{9}+\frac{14\!\cdots\!72}{5}a^{8}+\frac{58\!\cdots\!48}{5}a^{7}-\frac{22\!\cdots\!28}{5}a^{6}-\frac{61\!\cdots\!72}{5}a^{5}-\frac{14\!\cdots\!51}{10}a^{4}-\frac{22\!\cdots\!29}{10}a^{3}+\frac{35\!\cdots\!79}{10}a^{2}+\frac{36\!\cdots\!68}{5}a+\frac{22\!\cdots\!01}{5}$, $\frac{47\!\cdots\!31}{960}a^{22}+\frac{89\!\cdots\!19}{960}a^{21}+\frac{61\!\cdots\!53}{480}a^{20}+\frac{25\!\cdots\!81}{240}a^{19}-\frac{52\!\cdots\!43}{20}a^{18}-\frac{26\!\cdots\!33}{80}a^{17}-\frac{24\!\cdots\!07}{30}a^{16}-\frac{41\!\cdots\!93}{30}a^{15}-\frac{12\!\cdots\!57}{80}a^{14}-\frac{53\!\cdots\!41}{60}a^{13}+\frac{96\!\cdots\!41}{60}a^{12}+\frac{25\!\cdots\!91}{40}a^{11}+\frac{52\!\cdots\!49}{40}a^{10}+\frac{96\!\cdots\!27}{5}a^{9}+\frac{92\!\cdots\!38}{5}a^{8}+\frac{16\!\cdots\!29}{10}a^{7}-\frac{20\!\cdots\!77}{5}a^{6}-\frac{56\!\cdots\!43}{5}a^{5}-\frac{10\!\cdots\!22}{5}a^{4}-\frac{25\!\cdots\!01}{10}a^{3}-\frac{88\!\cdots\!32}{5}a^{2}+\frac{75\!\cdots\!32}{5}a+\frac{94\!\cdots\!69}{5}$, $\frac{29\!\cdots\!45}{8}a^{22}-\frac{22\!\cdots\!63}{48}a^{21}+\frac{49\!\cdots\!33}{96}a^{20}-\frac{41\!\cdots\!27}{96}a^{19}+\frac{72\!\cdots\!19}{8}a^{18}+\frac{66\!\cdots\!69}{96}a^{17}-\frac{34\!\cdots\!73}{16}a^{16}+\frac{21\!\cdots\!69}{48}a^{15}-\frac{39\!\cdots\!25}{48}a^{14}+\frac{39\!\cdots\!89}{3}a^{13}-\frac{37\!\cdots\!31}{2}a^{12}+\frac{19\!\cdots\!65}{8}a^{11}-28\!\cdots\!11a^{10}+\frac{10\!\cdots\!09}{4}a^{9}-\frac{22\!\cdots\!27}{2}a^{8}-\frac{50\!\cdots\!43}{2}a^{7}+\frac{38\!\cdots\!85}{4}a^{6}-\frac{43\!\cdots\!23}{2}a^{5}+\frac{80\!\cdots\!69}{2}a^{4}-\frac{13\!\cdots\!17}{2}a^{3}+97\!\cdots\!05a^{2}-13\!\cdots\!48a-32\!\cdots\!01$, $\frac{90\!\cdots\!51}{320}a^{22}+\frac{71\!\cdots\!07}{160}a^{21}+\frac{33\!\cdots\!29}{480}a^{20}+\frac{66\!\cdots\!17}{60}a^{19}+\frac{83\!\cdots\!79}{480}a^{18}+\frac{13\!\cdots\!91}{480}a^{17}+\frac{25\!\cdots\!73}{60}a^{16}+\frac{40\!\cdots\!07}{60}a^{15}+\frac{63\!\cdots\!43}{60}a^{14}+\frac{13\!\cdots\!31}{80}a^{13}+\frac{15\!\cdots\!53}{60}a^{12}+\frac{20\!\cdots\!41}{5}a^{11}+\frac{32\!\cdots\!49}{5}a^{10}+\frac{20\!\cdots\!49}{20}a^{9}+\frac{80\!\cdots\!74}{5}a^{8}+\frac{12\!\cdots\!21}{5}a^{7}+\frac{79\!\cdots\!51}{20}a^{6}+\frac{31\!\cdots\!16}{5}a^{5}+\frac{49\!\cdots\!79}{5}a^{4}+\frac{77\!\cdots\!51}{5}a^{3}+\frac{24\!\cdots\!33}{10}a^{2}+\frac{19\!\cdots\!41}{5}a+\frac{11\!\cdots\!77}{5}$, $\frac{12\!\cdots\!71}{48}a^{22}-\frac{48\!\cdots\!01}{192}a^{21}+\frac{58\!\cdots\!15}{24}a^{20}-\frac{74\!\cdots\!01}{32}a^{19}+\frac{17\!\cdots\!91}{8}a^{18}-\frac{20\!\cdots\!63}{96}a^{17}+\frac{49\!\cdots\!79}{24}a^{16}-\frac{93\!\cdots\!49}{48}a^{15}+\frac{72\!\cdots\!71}{4}a^{14}-\frac{76\!\cdots\!07}{48}a^{13}+\frac{15\!\cdots\!65}{12}a^{12}-\frac{69\!\cdots\!51}{8}a^{11}+\frac{25\!\cdots\!75}{8}a^{10}+\frac{14\!\cdots\!57}{4}a^{9}-\frac{43\!\cdots\!29}{4}a^{8}+\frac{67\!\cdots\!91}{4}a^{7}-\frac{71\!\cdots\!13}{4}a^{6}+\frac{15\!\cdots\!73}{2}a^{5}+22\!\cdots\!53a^{4}-85\!\cdots\!45a^{3}+\frac{39\!\cdots\!99}{2}a^{2}-37\!\cdots\!59a-34\!\cdots\!33$, $\frac{16\!\cdots\!17}{160}a^{22}-\frac{36\!\cdots\!31}{480}a^{21}-\frac{23\!\cdots\!43}{160}a^{20}+\frac{42\!\cdots\!91}{120}a^{19}-\frac{11\!\cdots\!33}{160}a^{18}-\frac{89\!\cdots\!29}{120}a^{17}+\frac{63\!\cdots\!47}{60}a^{16}+\frac{11\!\cdots\!97}{240}a^{15}-\frac{12\!\cdots\!27}{40}a^{14}+\frac{38\!\cdots\!82}{15}a^{13}+\frac{64\!\cdots\!58}{15}a^{12}-\frac{22\!\cdots\!99}{20}a^{11}+\frac{63\!\cdots\!09}{20}a^{10}+\frac{44\!\cdots\!61}{20}a^{9}-\frac{17\!\cdots\!64}{5}a^{8}-\frac{24\!\cdots\!29}{20}a^{7}+\frac{47\!\cdots\!36}{5}a^{6}-\frac{85\!\cdots\!57}{10}a^{5}-\frac{61\!\cdots\!79}{5}a^{4}+\frac{17\!\cdots\!14}{5}a^{3}-\frac{62\!\cdots\!94}{5}a^{2}-\frac{33\!\cdots\!91}{5}a-\frac{12\!\cdots\!57}{5}$, $\frac{94\!\cdots\!77}{960}a^{22}-\frac{46\!\cdots\!09}{320}a^{21}+\frac{16\!\cdots\!31}{80}a^{20}-\frac{37\!\cdots\!69}{120}a^{19}+\frac{21\!\cdots\!71}{480}a^{18}-\frac{32\!\cdots\!11}{480}a^{17}+\frac{97\!\cdots\!17}{10}a^{16}-\frac{34\!\cdots\!43}{240}a^{15}+\frac{16\!\cdots\!61}{80}a^{14}-\frac{37\!\cdots\!59}{120}a^{13}+\frac{90\!\cdots\!49}{20}a^{12}-\frac{66\!\cdots\!07}{10}a^{11}+\frac{39\!\cdots\!53}{40}a^{10}-\frac{14\!\cdots\!57}{10}a^{9}+\frac{21\!\cdots\!47}{10}a^{8}-\frac{61\!\cdots\!29}{20}a^{7}+\frac{90\!\cdots\!89}{20}a^{6}-\frac{66\!\cdots\!27}{10}a^{5}+\frac{97\!\cdots\!97}{10}a^{4}-\frac{14\!\cdots\!77}{10}a^{3}+\frac{10\!\cdots\!51}{5}a^{2}-\frac{15\!\cdots\!61}{5}a-\frac{42\!\cdots\!37}{5}$, $\frac{47\!\cdots\!27}{192}a^{22}+\frac{28\!\cdots\!39}{3}a^{21}-\frac{26\!\cdots\!05}{24}a^{20}-\frac{11\!\cdots\!85}{48}a^{19}-\frac{86\!\cdots\!83}{96}a^{18}+\frac{50\!\cdots\!51}{96}a^{17}+\frac{39\!\cdots\!49}{8}a^{16}-\frac{83\!\cdots\!69}{8}a^{15}-\frac{20\!\cdots\!27}{12}a^{14}+\frac{19\!\cdots\!41}{12}a^{13}+\frac{20\!\cdots\!31}{4}a^{12}-\frac{48\!\cdots\!65}{4}a^{11}-\frac{51\!\cdots\!51}{4}a^{10}-\frac{14\!\cdots\!09}{4}a^{9}+\frac{57\!\cdots\!39}{2}a^{8}+\frac{91\!\cdots\!81}{4}a^{7}-\frac{23\!\cdots\!23}{4}a^{6}-85\!\cdots\!96a^{5}+\frac{20\!\cdots\!09}{2}a^{4}+26\!\cdots\!25a^{3}-10\!\cdots\!66a^{2}-69\!\cdots\!09a-44\!\cdots\!73$, $\frac{18\!\cdots\!39}{192}a^{22}+\frac{29\!\cdots\!91}{64}a^{21}+\frac{75\!\cdots\!37}{24}a^{20}-\frac{35\!\cdots\!05}{32}a^{19}-\frac{27\!\cdots\!57}{48}a^{18}+\frac{58\!\cdots\!29}{24}a^{17}+\frac{12\!\cdots\!07}{48}a^{16}-\frac{10\!\cdots\!39}{24}a^{15}-\frac{40\!\cdots\!97}{48}a^{14}+\frac{15\!\cdots\!61}{24}a^{13}+\frac{57\!\cdots\!27}{24}a^{12}-\frac{30\!\cdots\!89}{8}a^{11}-\frac{47\!\cdots\!19}{8}a^{10}-\frac{85\!\cdots\!61}{4}a^{9}+\frac{52\!\cdots\!55}{4}a^{8}+\frac{24\!\cdots\!53}{2}a^{7}-25\!\cdots\!60a^{6}-\frac{84\!\cdots\!11}{2}a^{5}+\frac{78\!\cdots\!57}{2}a^{4}+\frac{24\!\cdots\!53}{2}a^{3}-30\!\cdots\!02a^{2}-31\!\cdots\!70a-21\!\cdots\!47$, $\frac{25\!\cdots\!33}{480}a^{22}-\frac{23\!\cdots\!87}{320}a^{21}+\frac{47\!\cdots\!19}{40}a^{20}-\frac{47\!\cdots\!73}{30}a^{19}+\frac{61\!\cdots\!79}{240}a^{18}-\frac{54\!\cdots\!71}{160}a^{17}+\frac{13\!\cdots\!39}{240}a^{16}-\frac{18\!\cdots\!49}{240}a^{15}+\frac{45\!\cdots\!39}{40}a^{14}-\frac{40\!\cdots\!09}{240}a^{13}+\frac{35\!\cdots\!34}{15}a^{12}-\frac{14\!\cdots\!39}{40}a^{11}+\frac{19\!\cdots\!69}{40}a^{10}-\frac{16\!\cdots\!97}{20}a^{9}+\frac{20\!\cdots\!17}{20}a^{8}-\frac{36\!\cdots\!57}{20}a^{7}+\frac{44\!\cdots\!07}{20}a^{6}-\frac{19\!\cdots\!63}{5}a^{5}+\frac{24\!\cdots\!73}{5}a^{4}-\frac{39\!\cdots\!78}{5}a^{3}+\frac{11\!\cdots\!01}{10}a^{2}-\frac{78\!\cdots\!03}{5}a-\frac{22\!\cdots\!21}{5}$, $\frac{14\!\cdots\!31}{480}a^{22}-\frac{33\!\cdots\!21}{480}a^{21}+\frac{28\!\cdots\!27}{160}a^{20}+\frac{38\!\cdots\!17}{240}a^{19}-\frac{48\!\cdots\!79}{480}a^{18}-\frac{56\!\cdots\!37}{160}a^{17}+\frac{88\!\cdots\!93}{240}a^{16}+\frac{16\!\cdots\!87}{240}a^{15}-\frac{13\!\cdots\!41}{120}a^{14}-\frac{15\!\cdots\!69}{120}a^{13}+\frac{63\!\cdots\!79}{20}a^{12}+\frac{77\!\cdots\!77}{40}a^{11}-\frac{41\!\cdots\!54}{5}a^{10}-\frac{16\!\cdots\!07}{10}a^{9}+\frac{40\!\cdots\!69}{20}a^{8}-\frac{66\!\cdots\!49}{20}a^{7}-\frac{95\!\cdots\!21}{20}a^{6}+\frac{25\!\cdots\!63}{10}a^{5}+\frac{10\!\cdots\!67}{10}a^{4}-\frac{48\!\cdots\!06}{5}a^{3}-\frac{10\!\cdots\!24}{5}a^{2}+\frac{15\!\cdots\!44}{5}a+\frac{18\!\cdots\!73}{5}$, $\frac{43\!\cdots\!07}{960}a^{22}+\frac{42\!\cdots\!07}{240}a^{21}-\frac{21\!\cdots\!53}{160}a^{20}+\frac{92\!\cdots\!17}{240}a^{19}-\frac{33\!\cdots\!69}{480}a^{18}+\frac{31\!\cdots\!77}{240}a^{17}-\frac{43\!\cdots\!97}{240}a^{16}+\frac{34\!\cdots\!01}{120}a^{15}-\frac{64\!\cdots\!91}{20}a^{14}+\frac{15\!\cdots\!57}{40}a^{13}-\frac{56\!\cdots\!71}{20}a^{12}+\frac{10\!\cdots\!37}{40}a^{11}+\frac{10\!\cdots\!89}{20}a^{10}-\frac{18\!\cdots\!07}{10}a^{9}+\frac{64\!\cdots\!99}{20}a^{8}-\frac{13\!\cdots\!09}{20}a^{7}+\frac{88\!\cdots\!07}{10}a^{6}-\frac{15\!\cdots\!27}{10}a^{5}+\frac{80\!\cdots\!71}{5}a^{4}-\frac{11\!\cdots\!71}{5}a^{3}+\frac{76\!\cdots\!21}{5}a^{2}-\frac{47\!\cdots\!71}{5}a-\frac{42\!\cdots\!17}{5}$ 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:  \( 72619230064900000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 72619230064900000000000000000 \cdot 1}{2\cdot\sqrt{82329339077098951705214211893347498601984994186425215376052749950135566336}}\cr\approx \mathstrut & 3.06996200461698 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 13248*x - 12672)
 
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 - 13248*x - 12672, 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 - 13248*x - 12672);
 
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 - 13248*x - 12672);
 
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 ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ R ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $19{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ R ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $20{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$$26$
\(3\) 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$Trivial$[\ ]$
11.11.11.3$x^{11} + 66 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
11.11.11.4$x^{11} + 77 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
\(23\) Copy content Toggle raw display 23.23.24.1$x^{23} + 23 x^{2} + 23$$23$$1$$24$$C_{23}:C_{11}$$[12/11]_{11}$