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Properties

Label 33.33.225...481.1
Degree $33$
Signature $[33, 0]$
Discriminant $2.250\times 10^{75}$
Root discriminant \(192.04\)
Ramified primes $23,37$
Class number not computed
Class group not computed
Galois group $C_{33}$ (as 33T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591)
 
gp: K = bnfinit(y^33 - 8*y^32 - 136*y^31 + 1010*y^30 + 9089*y^29 - 56260*y^28 - 391707*y^27 + 1778015*y^26 + 11776892*y^25 - 33684927*y^24 - 250499393*y^23 + 352249669*y^22 + 3723230595*y^21 - 869647969*y^20 - 37516024451*y^19 - 27213591295*y^18 + 241140895308*y^17 + 392273336476*y^16 - 846902299600*y^15 - 2477173011886*y^14 + 614677458641*y^13 + 7779541229922*y^12 + 5903210349213*y^11 - 9094078083973*y^10 - 16773542913789*y^9 - 4693562989906*y^8 + 8584245508513*y^7 + 7302900918860*y^6 + 488836631314*y^5 - 1567611092278*y^4 - 601455629303*y^3 - 17378174655*y^2 + 24259321620*y + 3109630591, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591)
 

\( x^{33} - 8 x^{32} - 136 x^{31} + 1010 x^{30} + 9089 x^{29} - 56260 x^{28} - 391707 x^{27} + \cdots + 3109630591 \) Copy content Toggle raw display

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

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[33, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2250468870721257864915422491944078011918280366020799086767808384626842687481\) \(\medspace = 23^{30}\cdot 37^{22}\) 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:  \(192.04\)
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:  $23^{10/11}37^{2/3}\approx 192.04464972398003$
Ramified primes:   \(23\), \(37\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $33$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(851=23\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{851}(1,·)$, $\chi_{851}(519,·)$, $\chi_{851}(269,·)$, $\chi_{851}(26,·)$, $\chi_{851}(285,·)$, $\chi_{851}(417,·)$, $\chi_{851}(676,·)$, $\chi_{851}(556,·)$, $\chi_{851}(174,·)$, $\chi_{851}(47,·)$, $\chi_{851}(692,·)$, $\chi_{851}(565,·)$, $\chi_{851}(186,·)$, $\chi_{851}(445,·)$, $\chi_{851}(581,·)$, $\chi_{851}(840,·)$, $\chi_{851}(75,·)$, $\chi_{851}(334,·)$, $\chi_{851}(593,·)$, $\chi_{851}(211,·)$, $\chi_{851}(729,·)$, $\chi_{851}(602,·)$, $\chi_{851}(223,·)$, $\chi_{851}(100,·)$, $\chi_{851}(232,·)$, $\chi_{851}(491,·)$, $\chi_{851}(371,·)$, $\chi_{851}(630,·)$, $\chi_{851}(248,·)$, $\chi_{851}(121,·)$, $\chi_{851}(507,·)$, $\chi_{851}(380,·)$, $\chi_{851}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{47}a^{25}-\frac{17}{47}a^{24}-\frac{2}{47}a^{23}+\frac{2}{47}a^{22}-\frac{3}{47}a^{21}+\frac{16}{47}a^{20}+\frac{9}{47}a^{19}+\frac{11}{47}a^{18}-\frac{6}{47}a^{17}-\frac{13}{47}a^{16}-\frac{5}{47}a^{15}+\frac{1}{47}a^{14}+\frac{13}{47}a^{13}+\frac{13}{47}a^{12}+\frac{7}{47}a^{11}-\frac{22}{47}a^{10}+\frac{2}{47}a^{9}+\frac{14}{47}a^{8}+\frac{21}{47}a^{7}+\frac{14}{47}a^{6}+\frac{13}{47}a^{5}+\frac{23}{47}a^{4}-\frac{7}{47}a^{3}+\frac{5}{47}a^{2}+\frac{4}{47}a$, $\frac{1}{47}a^{26}-\frac{9}{47}a^{24}+\frac{15}{47}a^{23}-\frac{16}{47}a^{22}+\frac{12}{47}a^{21}-\frac{1}{47}a^{20}+\frac{23}{47}a^{19}-\frac{7}{47}a^{18}-\frac{21}{47}a^{17}+\frac{9}{47}a^{16}+\frac{10}{47}a^{15}-\frac{17}{47}a^{14}-\frac{1}{47}a^{13}-\frac{7}{47}a^{12}+\frac{3}{47}a^{11}+\frac{4}{47}a^{10}+\frac{1}{47}a^{9}-\frac{23}{47}a^{8}-\frac{5}{47}a^{7}+\frac{16}{47}a^{6}+\frac{9}{47}a^{5}+\frac{8}{47}a^{4}-\frac{20}{47}a^{3}-\frac{5}{47}a^{2}+\frac{21}{47}a$, $\frac{1}{47}a^{27}+\frac{3}{47}a^{24}+\frac{13}{47}a^{23}-\frac{17}{47}a^{22}+\frac{19}{47}a^{21}-\frac{21}{47}a^{20}-\frac{20}{47}a^{19}-\frac{16}{47}a^{18}+\frac{2}{47}a^{17}-\frac{13}{47}a^{16}-\frac{15}{47}a^{15}+\frac{8}{47}a^{14}+\frac{16}{47}a^{13}-\frac{21}{47}a^{12}+\frac{20}{47}a^{11}-\frac{9}{47}a^{10}-\frac{5}{47}a^{9}-\frac{20}{47}a^{8}+\frac{17}{47}a^{7}-\frac{6}{47}a^{6}-\frac{16}{47}a^{5}-\frac{1}{47}a^{4}-\frac{21}{47}a^{3}+\frac{19}{47}a^{2}-\frac{11}{47}a$, $\frac{1}{47}a^{28}+\frac{17}{47}a^{24}-\frac{11}{47}a^{23}+\frac{13}{47}a^{22}-\frac{12}{47}a^{21}-\frac{21}{47}a^{20}+\frac{4}{47}a^{19}+\frac{16}{47}a^{18}+\frac{5}{47}a^{17}-\frac{23}{47}a^{16}+\frac{23}{47}a^{15}+\frac{13}{47}a^{14}-\frac{13}{47}a^{13}-\frac{19}{47}a^{12}+\frac{17}{47}a^{11}+\frac{14}{47}a^{10}+\frac{21}{47}a^{9}+\frac{22}{47}a^{8}-\frac{22}{47}a^{7}-\frac{11}{47}a^{6}+\frac{7}{47}a^{5}+\frac{4}{47}a^{4}-\frac{7}{47}a^{3}+\frac{21}{47}a^{2}-\frac{12}{47}a$, $\frac{1}{47}a^{29}-\frac{4}{47}a^{24}+\frac{1}{47}a^{22}-\frac{17}{47}a^{21}+\frac{14}{47}a^{20}+\frac{4}{47}a^{19}+\frac{6}{47}a^{18}-\frac{15}{47}a^{17}+\frac{9}{47}a^{16}+\frac{4}{47}a^{15}+\frac{17}{47}a^{14}-\frac{5}{47}a^{13}-\frac{16}{47}a^{12}-\frac{11}{47}a^{11}+\frac{19}{47}a^{10}-\frac{12}{47}a^{9}+\frac{22}{47}a^{8}+\frac{8}{47}a^{7}+\frac{4}{47}a^{6}+\frac{18}{47}a^{5}-\frac{22}{47}a^{4}-\frac{1}{47}a^{3}-\frac{3}{47}a^{2}-\frac{21}{47}a$, $\frac{1}{47}a^{30}-\frac{21}{47}a^{24}-\frac{7}{47}a^{23}-\frac{9}{47}a^{22}+\frac{2}{47}a^{21}+\frac{21}{47}a^{20}-\frac{5}{47}a^{19}-\frac{18}{47}a^{18}-\frac{15}{47}a^{17}-\frac{1}{47}a^{16}-\frac{3}{47}a^{15}-\frac{1}{47}a^{14}-\frac{11}{47}a^{13}-\frac{6}{47}a^{12}-\frac{6}{47}a^{10}-\frac{17}{47}a^{9}+\frac{17}{47}a^{8}-\frac{6}{47}a^{7}-\frac{20}{47}a^{6}-\frac{17}{47}a^{5}-\frac{3}{47}a^{4}+\frac{16}{47}a^{3}-\frac{1}{47}a^{2}+\frac{16}{47}a$, $\frac{1}{2209}a^{31}+\frac{7}{2209}a^{30}+\frac{19}{2209}a^{29}-\frac{19}{2209}a^{27}-\frac{4}{2209}a^{26}+\frac{15}{2209}a^{25}+\frac{218}{2209}a^{24}+\frac{268}{2209}a^{23}-\frac{476}{2209}a^{22}+\frac{417}{2209}a^{21}-\frac{869}{2209}a^{20}+\frac{24}{2209}a^{19}-\frac{1038}{2209}a^{18}+\frac{990}{2209}a^{17}+\frac{139}{2209}a^{16}-\frac{539}{2209}a^{15}+\frac{774}{2209}a^{14}+\frac{883}{2209}a^{13}+\frac{361}{2209}a^{12}+\frac{397}{2209}a^{11}-\frac{1087}{2209}a^{10}-\frac{1013}{2209}a^{9}+\frac{50}{2209}a^{8}+\frac{402}{2209}a^{7}+\frac{50}{2209}a^{6}-\frac{454}{2209}a^{5}-\frac{595}{2209}a^{4}-\frac{198}{2209}a^{3}-\frac{632}{2209}a^{2}-\frac{817}{2209}a-\frac{10}{47}$, $\frac{1}{18\!\cdots\!67}a^{32}-\frac{39\!\cdots\!93}{18\!\cdots\!67}a^{31}+\frac{17\!\cdots\!62}{18\!\cdots\!67}a^{30}+\frac{15\!\cdots\!85}{18\!\cdots\!67}a^{29}-\frac{36\!\cdots\!26}{18\!\cdots\!67}a^{28}+\frac{78\!\cdots\!86}{18\!\cdots\!67}a^{27}+\frac{14\!\cdots\!02}{18\!\cdots\!67}a^{26}-\frac{12\!\cdots\!07}{18\!\cdots\!67}a^{25}-\frac{52\!\cdots\!68}{18\!\cdots\!67}a^{24}+\frac{65\!\cdots\!57}{18\!\cdots\!67}a^{23}+\frac{74\!\cdots\!55}{18\!\cdots\!67}a^{22}+\frac{89\!\cdots\!55}{18\!\cdots\!67}a^{21}-\frac{55\!\cdots\!94}{18\!\cdots\!67}a^{20}+\frac{30\!\cdots\!08}{18\!\cdots\!67}a^{19}-\frac{40\!\cdots\!40}{13\!\cdots\!91}a^{18}-\frac{53\!\cdots\!88}{18\!\cdots\!67}a^{17}-\frac{51\!\cdots\!09}{18\!\cdots\!67}a^{16}-\frac{70\!\cdots\!19}{18\!\cdots\!67}a^{15}+\frac{22\!\cdots\!77}{18\!\cdots\!67}a^{14}+\frac{10\!\cdots\!11}{18\!\cdots\!67}a^{13}+\frac{54\!\cdots\!73}{18\!\cdots\!67}a^{12}+\frac{47\!\cdots\!10}{18\!\cdots\!67}a^{11}+\frac{53\!\cdots\!35}{18\!\cdots\!67}a^{10}+\frac{40\!\cdots\!88}{18\!\cdots\!67}a^{9}+\frac{35\!\cdots\!69}{18\!\cdots\!67}a^{8}+\frac{67\!\cdots\!14}{18\!\cdots\!67}a^{7}-\frac{31\!\cdots\!38}{18\!\cdots\!67}a^{6}-\frac{11\!\cdots\!68}{18\!\cdots\!67}a^{5}-\frac{41\!\cdots\!61}{18\!\cdots\!67}a^{4}-\frac{74\!\cdots\!30}{18\!\cdots\!67}a^{3}+\frac{13\!\cdots\!66}{18\!\cdots\!67}a^{2}-\frac{24\!\cdots\!35}{18\!\cdots\!67}a-\frac{14\!\cdots\!32}{38\!\cdots\!61}$ 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

not computed

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:  $32$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591)
 
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^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591, 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^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591);
 
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^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591);
 
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

$C_{33}$ (as 33T1):

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 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

3.3.1369.1, \(\Q(\zeta_{23})^+\)

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 $33$ $33$ $33$ $33$ ${\href{/padicField/11.11.0.1}{11} }^{3}$ $33$ $33$ $33$ R ${\href{/padicField/29.11.0.1}{11} }^{3}$ ${\href{/padicField/31.11.0.1}{11} }^{3}$ R $33$ ${\href{/padicField/43.11.0.1}{11} }^{3}$ ${\href{/padicField/47.1.0.1}{1} }^{33}$ $33$ $33$

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
\(23\) Copy content Toggle raw display 23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
\(37\) Copy content Toggle raw display Deg $33$$3$$11$$22$