Properties

Label 18.0.680...571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.801\times 10^{29}$
Root discriminant \(45.43\)
Ramified primes $11,19$
Class number $1084$ (GRH)
Class group [2, 542] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641)
 
gp: K = bnfinit(y^18 - 7*y^17 + 31*y^16 - 98*y^15 + 331*y^14 - 957*y^13 + 2693*y^12 - 6227*y^11 + 15226*y^10 - 30674*y^9 + 66392*y^8 - 111391*y^7 + 216508*y^6 - 293668*y^5 + 517517*y^4 - 511304*y^3 + 833369*y^2 - 453934*y + 713641, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641)
 

\( x^{18} - 7 x^{17} + 31 x^{16} - 98 x^{15} + 331 x^{14} - 957 x^{13} + 2693 x^{12} - 6227 x^{11} + 15226 x^{10} - 30674 x^{9} + 66392 x^{8} - 111391 x^{7} + 216508 x^{6} + \cdots + 713641 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-680129765110548404696137774571\) \(\medspace = -\,11^{9}\cdot 19^{16}\) 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:  \(45.43\)
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:  $11^{1/2}19^{8/9}\approx 45.432455522435575$
Ramified primes:   \(11\), \(19\) 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:  \(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(131,·)$, $\chi_{209}(197,·)$, $\chi_{209}(199,·)$, $\chi_{209}(23,·)$, $\chi_{209}(142,·)$, $\chi_{209}(144,·)$, $\chi_{209}(87,·)$, $\chi_{209}(153,·)$, $\chi_{209}(111,·)$, $\chi_{209}(100,·)$, $\chi_{209}(43,·)$, $\chi_{209}(45,·)$, $\chi_{209}(175,·)$, $\chi_{209}(177,·)$, $\chi_{209}(54,·)$, $\chi_{209}(120,·)$, $\chi_{209}(188,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37}a^{16}-\frac{12}{37}a^{15}+\frac{8}{37}a^{14}+\frac{7}{37}a^{13}+\frac{2}{37}a^{12}+\frac{6}{37}a^{11}+\frac{18}{37}a^{10}-\frac{7}{37}a^{9}+\frac{3}{37}a^{8}+\frac{10}{37}a^{7}+\frac{11}{37}a^{6}-\frac{18}{37}a^{5}+\frac{12}{37}a^{4}-\frac{8}{37}a^{3}+\frac{4}{37}a^{2}+\frac{14}{37}a-\frac{14}{37}$, $\frac{1}{28\!\cdots\!63}a^{17}-\frac{23\!\cdots\!74}{28\!\cdots\!63}a^{16}-\frac{12\!\cdots\!37}{28\!\cdots\!63}a^{15}+\frac{82\!\cdots\!81}{28\!\cdots\!63}a^{14}+\frac{11\!\cdots\!91}{28\!\cdots\!63}a^{13}-\frac{56\!\cdots\!75}{28\!\cdots\!63}a^{12}-\frac{12\!\cdots\!99}{28\!\cdots\!63}a^{11}-\frac{10\!\cdots\!65}{28\!\cdots\!63}a^{10}-\frac{96\!\cdots\!88}{28\!\cdots\!63}a^{9}-\frac{59\!\cdots\!07}{28\!\cdots\!63}a^{8}+\frac{99\!\cdots\!83}{28\!\cdots\!63}a^{7}+\frac{76\!\cdots\!56}{28\!\cdots\!63}a^{6}-\frac{96\!\cdots\!68}{28\!\cdots\!63}a^{5}+\frac{72\!\cdots\!59}{28\!\cdots\!63}a^{4}+\frac{48\!\cdots\!78}{28\!\cdots\!63}a^{3}+\frac{10\!\cdots\!38}{28\!\cdots\!63}a^{2}-\frac{11\!\cdots\!41}{28\!\cdots\!63}a-\frac{12\!\cdots\!02}{28\!\cdots\!63}$ 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

$C_{2}\times C_{542}$, which has order $1084$ (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:  $8$
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:   $\frac{20\!\cdots\!38}{28\!\cdots\!63}a^{17}-\frac{54\!\cdots\!26}{76\!\cdots\!99}a^{16}+\frac{96\!\cdots\!46}{28\!\cdots\!63}a^{15}-\frac{32\!\cdots\!91}{28\!\cdots\!63}a^{14}+\frac{99\!\cdots\!15}{28\!\cdots\!63}a^{13}-\frac{30\!\cdots\!90}{28\!\cdots\!63}a^{12}+\frac{81\!\cdots\!21}{28\!\cdots\!63}a^{11}-\frac{19\!\cdots\!09}{28\!\cdots\!63}a^{10}+\frac{42\!\cdots\!66}{28\!\cdots\!63}a^{9}-\frac{91\!\cdots\!56}{28\!\cdots\!63}a^{8}+\frac{17\!\cdots\!73}{28\!\cdots\!63}a^{7}-\frac{31\!\cdots\!29}{28\!\cdots\!63}a^{6}+\frac{47\!\cdots\!24}{28\!\cdots\!63}a^{5}-\frac{75\!\cdots\!40}{28\!\cdots\!63}a^{4}+\frac{91\!\cdots\!81}{28\!\cdots\!63}a^{3}-\frac{11\!\cdots\!01}{28\!\cdots\!63}a^{2}+\frac{84\!\cdots\!24}{28\!\cdots\!63}a-\frac{92\!\cdots\!13}{28\!\cdots\!63}$, $\frac{39\!\cdots\!52}{28\!\cdots\!63}a^{17}-\frac{68\!\cdots\!54}{28\!\cdots\!63}a^{16}+\frac{45\!\cdots\!10}{28\!\cdots\!63}a^{15}-\frac{20\!\cdots\!65}{28\!\cdots\!63}a^{14}+\frac{69\!\cdots\!19}{28\!\cdots\!63}a^{13}-\frac{20\!\cdots\!28}{28\!\cdots\!63}a^{12}+\frac{54\!\cdots\!33}{28\!\cdots\!63}a^{11}-\frac{14\!\cdots\!65}{28\!\cdots\!63}a^{10}+\frac{32\!\cdots\!62}{28\!\cdots\!63}a^{9}-\frac{71\!\cdots\!93}{28\!\cdots\!63}a^{8}+\frac{13\!\cdots\!15}{28\!\cdots\!63}a^{7}-\frac{25\!\cdots\!15}{28\!\cdots\!63}a^{6}+\frac{37\!\cdots\!08}{28\!\cdots\!63}a^{5}-\frac{63\!\cdots\!50}{28\!\cdots\!63}a^{4}+\frac{66\!\cdots\!43}{28\!\cdots\!63}a^{3}-\frac{10\!\cdots\!47}{28\!\cdots\!63}a^{2}+\frac{57\!\cdots\!13}{28\!\cdots\!63}a-\frac{93\!\cdots\!84}{28\!\cdots\!63}$, $\frac{42\!\cdots\!42}{28\!\cdots\!63}a^{17}-\frac{28\!\cdots\!87}{28\!\cdots\!63}a^{16}+\frac{12\!\cdots\!26}{28\!\cdots\!63}a^{15}-\frac{38\!\cdots\!40}{28\!\cdots\!63}a^{14}+\frac{13\!\cdots\!56}{28\!\cdots\!63}a^{13}-\frac{37\!\cdots\!55}{28\!\cdots\!63}a^{12}+\frac{10\!\cdots\!40}{28\!\cdots\!63}a^{11}-\frac{24\!\cdots\!59}{28\!\cdots\!63}a^{10}+\frac{61\!\cdots\!40}{28\!\cdots\!63}a^{9}-\frac{12\!\cdots\!72}{28\!\cdots\!63}a^{8}+\frac{27\!\cdots\!60}{28\!\cdots\!63}a^{7}-\frac{44\!\cdots\!42}{28\!\cdots\!63}a^{6}+\frac{92\!\cdots\!95}{28\!\cdots\!63}a^{5}-\frac{11\!\cdots\!41}{28\!\cdots\!63}a^{4}+\frac{24\!\cdots\!39}{28\!\cdots\!63}a^{3}-\frac{20\!\cdots\!76}{28\!\cdots\!63}a^{2}+\frac{29\!\cdots\!12}{28\!\cdots\!63}a-\frac{42\!\cdots\!30}{28\!\cdots\!63}$, $\frac{12\!\cdots\!28}{28\!\cdots\!63}a^{17}-\frac{12\!\cdots\!35}{28\!\cdots\!63}a^{16}+\frac{63\!\cdots\!74}{28\!\cdots\!63}a^{15}-\frac{23\!\cdots\!48}{28\!\cdots\!63}a^{14}+\frac{76\!\cdots\!78}{28\!\cdots\!63}a^{13}-\frac{24\!\cdots\!23}{28\!\cdots\!63}a^{12}+\frac{69\!\cdots\!90}{28\!\cdots\!63}a^{11}-\frac{18\!\cdots\!55}{28\!\cdots\!63}a^{10}+\frac{43\!\cdots\!68}{28\!\cdots\!63}a^{9}-\frac{10\!\cdots\!64}{28\!\cdots\!63}a^{8}+\frac{20\!\cdots\!00}{28\!\cdots\!63}a^{7}-\frac{45\!\cdots\!60}{28\!\cdots\!63}a^{6}+\frac{76\!\cdots\!65}{28\!\cdots\!63}a^{5}-\frac{16\!\cdots\!91}{28\!\cdots\!63}a^{4}+\frac{21\!\cdots\!55}{28\!\cdots\!63}a^{3}-\frac{37\!\cdots\!67}{28\!\cdots\!63}a^{2}+\frac{28\!\cdots\!79}{28\!\cdots\!63}a-\frac{85\!\cdots\!01}{28\!\cdots\!63}$, $\frac{13\!\cdots\!56}{28\!\cdots\!63}a^{17}-\frac{10\!\cdots\!81}{28\!\cdots\!63}a^{16}+\frac{54\!\cdots\!66}{28\!\cdots\!63}a^{15}-\frac{21\!\cdots\!93}{28\!\cdots\!63}a^{14}+\frac{73\!\cdots\!95}{28\!\cdots\!63}a^{13}-\frac{20\!\cdots\!82}{28\!\cdots\!63}a^{12}+\frac{53\!\cdots\!09}{28\!\cdots\!63}a^{11}-\frac{12\!\cdots\!19}{28\!\cdots\!63}a^{10}+\frac{30\!\cdots\!76}{28\!\cdots\!63}a^{9}-\frac{57\!\cdots\!93}{28\!\cdots\!63}a^{8}+\frac{11\!\cdots\!75}{28\!\cdots\!63}a^{7}-\frac{17\!\cdots\!17}{28\!\cdots\!63}a^{6}+\frac{29\!\cdots\!68}{28\!\cdots\!63}a^{5}-\frac{29\!\cdots\!85}{28\!\cdots\!63}a^{4}+\frac{45\!\cdots\!85}{28\!\cdots\!63}a^{3}-\frac{18\!\cdots\!37}{28\!\cdots\!63}a^{2}+\frac{32\!\cdots\!43}{28\!\cdots\!63}a+\frac{46\!\cdots\!78}{28\!\cdots\!63}$, $\frac{45\!\cdots\!76}{28\!\cdots\!63}a^{17}-\frac{30\!\cdots\!58}{28\!\cdots\!63}a^{16}+\frac{11\!\cdots\!56}{28\!\cdots\!63}a^{15}-\frac{29\!\cdots\!61}{28\!\cdots\!63}a^{14}+\frac{89\!\cdots\!81}{28\!\cdots\!63}a^{13}-\frac{25\!\cdots\!72}{28\!\cdots\!63}a^{12}+\frac{67\!\cdots\!73}{28\!\cdots\!63}a^{11}-\frac{12\!\cdots\!69}{28\!\cdots\!63}a^{10}+\frac{27\!\cdots\!74}{28\!\cdots\!63}a^{9}-\frac{47\!\cdots\!53}{28\!\cdots\!63}a^{8}+\frac{10\!\cdots\!13}{28\!\cdots\!63}a^{7}-\frac{11\!\cdots\!34}{28\!\cdots\!63}a^{6}+\frac{22\!\cdots\!93}{28\!\cdots\!63}a^{5}-\frac{12\!\cdots\!15}{28\!\cdots\!63}a^{4}+\frac{35\!\cdots\!58}{28\!\cdots\!63}a^{3}+\frac{98\!\cdots\!06}{28\!\cdots\!63}a^{2}+\frac{64\!\cdots\!92}{76\!\cdots\!99}a+\frac{55\!\cdots\!56}{28\!\cdots\!63}$, $\frac{18\!\cdots\!44}{28\!\cdots\!63}a^{17}-\frac{82\!\cdots\!53}{28\!\cdots\!63}a^{16}+\frac{14\!\cdots\!66}{28\!\cdots\!63}a^{15}+\frac{13\!\cdots\!35}{28\!\cdots\!63}a^{14}-\frac{12\!\cdots\!37}{28\!\cdots\!63}a^{13}+\frac{12\!\cdots\!22}{28\!\cdots\!63}a^{12}-\frac{57\!\cdots\!15}{28\!\cdots\!63}a^{11}+\frac{28\!\cdots\!22}{28\!\cdots\!63}a^{10}-\frac{53\!\cdots\!71}{28\!\cdots\!63}a^{9}+\frac{16\!\cdots\!99}{28\!\cdots\!63}a^{8}-\frac{27\!\cdots\!57}{28\!\cdots\!63}a^{7}+\frac{62\!\cdots\!13}{28\!\cdots\!63}a^{6}-\frac{73\!\cdots\!10}{28\!\cdots\!63}a^{5}+\frac{16\!\cdots\!60}{28\!\cdots\!63}a^{4}-\frac{94\!\cdots\!94}{28\!\cdots\!63}a^{3}+\frac{19\!\cdots\!25}{28\!\cdots\!63}a^{2}-\frac{36\!\cdots\!77}{28\!\cdots\!63}a-\frac{30\!\cdots\!90}{28\!\cdots\!63}$, $\frac{15\!\cdots\!26}{28\!\cdots\!63}a^{17}-\frac{10\!\cdots\!70}{28\!\cdots\!63}a^{16}+\frac{34\!\cdots\!48}{28\!\cdots\!63}a^{15}-\frac{72\!\cdots\!61}{28\!\cdots\!63}a^{14}+\frac{21\!\cdots\!35}{28\!\cdots\!63}a^{13}-\frac{64\!\cdots\!34}{28\!\cdots\!63}a^{12}+\frac{15\!\cdots\!67}{28\!\cdots\!63}a^{11}-\frac{27\!\cdots\!02}{28\!\cdots\!63}a^{10}+\frac{56\!\cdots\!31}{28\!\cdots\!63}a^{9}-\frac{91\!\cdots\!31}{28\!\cdots\!63}a^{8}+\frac{18\!\cdots\!61}{28\!\cdots\!63}a^{7}-\frac{17\!\cdots\!96}{28\!\cdots\!63}a^{6}+\frac{34\!\cdots\!37}{28\!\cdots\!63}a^{5}+\frac{22\!\cdots\!42}{28\!\cdots\!63}a^{4}+\frac{48\!\cdots\!15}{28\!\cdots\!63}a^{3}+\frac{14\!\cdots\!17}{28\!\cdots\!63}a^{2}+\frac{15\!\cdots\!93}{28\!\cdots\!63}a+\frac{52\!\cdots\!05}{28\!\cdots\!63}$ 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:  \( 22305.8950792 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 1084}{2\cdot\sqrt{680129765110548404696137774571}}\cr\approx \mathstrut & 0.223739089291 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641)
 
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^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641, 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^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641);
 
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^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.361.1, 6.0.173457251.1, \(\Q(\zeta_{19})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 $18$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ R $18$ $18$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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
\(11\) Copy content Toggle raw display 11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(19\) Copy content Toggle raw display 19.18.16.1$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$$9$$2$$16$$C_{18}$$[\ ]_{9}^{2}$