Properties

Label 21.1.156...857.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.561\times 10^{26}$
Root discriminant \(17.67\)
Ramified primes $23,71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1)
 
gp: K = bnfinit(y^21 - y^20 - y^19 + 2*y^18 + 5*y^17 + 2*y^16 - 10*y^15 - 2*y^14 + 23*y^13 + 19*y^12 - 11*y^11 - 28*y^10 + 8*y^9 + 47*y^8 + 23*y^7 - 13*y^6 - 24*y^5 - 7*y^4 + 11*y^3 + 6*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1)
 

\( x^{21} - x^{20} - x^{19} + 2 x^{18} + 5 x^{17} + 2 x^{16} - 10 x^{15} - 2 x^{14} + 23 x^{13} + 19 x^{12} - 11 x^{11} - 28 x^{10} + 8 x^{9} + 47 x^{8} + 23 x^{7} - 13 x^{6} - 24 x^{5} - 7 x^{4} + 11 x^{3} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(156106141952973043121291857\) \(\medspace = 23^{7}\cdot 71^{9}\) 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:  \(17.67\)
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:  $23^{1/2}71^{1/2}\approx 40.4103947023535$
Ramified primes:   \(23\), \(71\) 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:  \(\Q(\sqrt{1633}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{3}{7}a^{15}-\frac{1}{7}a^{14}-\frac{3}{7}a^{13}+\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{16}+\frac{2}{7}a^{15}-\frac{2}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{8}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{19}-\frac{1}{7}a^{16}+\frac{2}{7}a^{14}-\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{4}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{67050907}a^{20}+\frac{619012}{9578701}a^{19}-\frac{2430104}{67050907}a^{18}-\frac{48420}{9578701}a^{17}+\frac{1154233}{9578701}a^{16}+\frac{25935584}{67050907}a^{15}+\frac{2144834}{9578701}a^{14}-\frac{21230934}{67050907}a^{13}+\frac{21553102}{67050907}a^{12}-\frac{1793716}{67050907}a^{11}+\frac{25671549}{67050907}a^{10}-\frac{19772818}{67050907}a^{9}+\frac{31783945}{67050907}a^{8}-\frac{1214318}{9578701}a^{7}-\frac{17854070}{67050907}a^{6}+\frac{28224926}{67050907}a^{5}-\frac{23307932}{67050907}a^{4}+\frac{24879307}{67050907}a^{3}-\frac{28098866}{67050907}a^{2}+\frac{7421087}{67050907}a-\frac{6164758}{67050907}$ 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:  $10$
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:   $\frac{12008369}{67050907}a^{20}-\frac{33047792}{67050907}a^{19}+\frac{22843134}{67050907}a^{18}+\frac{26725256}{67050907}a^{17}+\frac{10752985}{67050907}a^{16}-\frac{53248837}{67050907}a^{15}-\frac{97689134}{67050907}a^{14}+\frac{191647202}{67050907}a^{13}+\frac{24536663}{9578701}a^{12}-\frac{227716730}{67050907}a^{11}-\frac{219135061}{67050907}a^{10}+\frac{9267503}{9578701}a^{9}+\frac{448525808}{67050907}a^{8}+\frac{72959049}{67050907}a^{7}-\frac{447223565}{67050907}a^{6}-\frac{73499782}{67050907}a^{5}+\frac{56067461}{67050907}a^{4}+\frac{166836778}{67050907}a^{3}+\frac{49747284}{67050907}a^{2}-\frac{140417181}{67050907}a-\frac{14278335}{67050907}$, $\frac{8887528}{67050907}a^{20}-\frac{5276769}{67050907}a^{19}-\frac{421194}{9578701}a^{18}-\frac{113737}{67050907}a^{17}+\frac{55672847}{67050907}a^{16}+\frac{46271614}{67050907}a^{15}-\frac{44987521}{67050907}a^{14}-\frac{6257727}{9578701}a^{13}+\frac{132251481}{67050907}a^{12}+\frac{274580767}{67050907}a^{11}+\frac{123352825}{67050907}a^{10}-\frac{30454823}{9578701}a^{9}-\frac{38174746}{67050907}a^{8}+\frac{46769497}{9578701}a^{7}+\frac{451246056}{67050907}a^{6}+\frac{192928256}{67050907}a^{5}-\frac{144150016}{67050907}a^{4}-\frac{48805293}{67050907}a^{3}+\frac{2342726}{9578701}a^{2}+\frac{14897336}{67050907}a+\frac{49451407}{67050907}$, $\frac{22489672}{67050907}a^{20}-\frac{33432431}{67050907}a^{19}+\frac{4118413}{67050907}a^{18}+\frac{4704945}{9578701}a^{17}+\frac{13135473}{9578701}a^{16}+\frac{11894625}{67050907}a^{15}-\frac{178824222}{67050907}a^{14}+\frac{75382215}{67050907}a^{13}+\frac{57463174}{9578701}a^{12}+\frac{28982315}{9578701}a^{11}-\frac{154429789}{67050907}a^{10}-\frac{325116961}{67050907}a^{9}+\frac{338096482}{67050907}a^{8}+\frac{617147831}{67050907}a^{7}+\frac{214770755}{67050907}a^{6}+\frac{5561692}{9578701}a^{5}-\frac{201381921}{67050907}a^{4}-\frac{94581715}{67050907}a^{3}-\frac{8593195}{67050907}a^{2}+\frac{20566820}{67050907}a+\frac{64334063}{67050907}$, $\frac{7278451}{67050907}a^{20}-\frac{20830646}{67050907}a^{19}+\frac{13762131}{67050907}a^{18}+\frac{17103208}{67050907}a^{17}+\frac{8773211}{67050907}a^{16}-\frac{41217677}{67050907}a^{15}-\frac{62917751}{67050907}a^{14}+\frac{119394070}{67050907}a^{13}+\frac{118913021}{67050907}a^{12}-\frac{145171162}{67050907}a^{11}-\frac{175926347}{67050907}a^{10}+\frac{41484117}{67050907}a^{9}+\frac{302900347}{67050907}a^{8}+\frac{9138292}{9578701}a^{7}-\frac{277482844}{67050907}a^{6}-\frac{65597912}{67050907}a^{5}+\frac{31243986}{67050907}a^{4}+\frac{108651232}{67050907}a^{3}+\frac{33783481}{67050907}a^{2}+\frac{939566}{67050907}a+\frac{22369884}{67050907}$, $\frac{16040933}{67050907}a^{20}-\frac{41707214}{67050907}a^{19}+\frac{10888033}{67050907}a^{18}+\frac{6152127}{9578701}a^{17}+\frac{33238605}{67050907}a^{16}-\frac{75185357}{67050907}a^{15}-\frac{32488419}{9578701}a^{14}+\frac{153244692}{67050907}a^{13}+\frac{338544386}{67050907}a^{12}-\frac{26362744}{9578701}a^{11}-\frac{565116098}{67050907}a^{10}-\frac{430299269}{67050907}a^{9}+\frac{66111854}{9578701}a^{8}+\frac{477109361}{67050907}a^{7}-\frac{520337718}{67050907}a^{6}-\frac{713627975}{67050907}a^{5}-\frac{616159086}{67050907}a^{4}-\frac{37045104}{67050907}a^{3}+\frac{204717496}{67050907}a^{2}-\frac{15170024}{67050907}a+\frac{46714978}{67050907}$, $\frac{2704416}{6095537}a^{20}-\frac{4418462}{6095537}a^{19}-\frac{944630}{6095537}a^{18}+\frac{6870274}{6095537}a^{17}+\frac{10995805}{6095537}a^{16}-\frac{4455616}{6095537}a^{15}-\frac{29238339}{6095537}a^{14}+\frac{11090250}{6095537}a^{13}+\frac{66673450}{6095537}a^{12}+\frac{13421304}{6095537}a^{11}-\frac{64207681}{6095537}a^{10}-\frac{55510962}{6095537}a^{9}+\frac{73380672}{6095537}a^{8}+\frac{113457210}{6095537}a^{7}-\frac{13255122}{6095537}a^{6}-\frac{77185698}{6095537}a^{5}-\frac{39086402}{6095537}a^{4}+\frac{26062517}{6095537}a^{3}+\frac{6373848}{870791}a^{2}+\frac{1071640}{6095537}a-\frac{10180248}{6095537}$, $\frac{6140432}{6095537}a^{20}-\frac{8740339}{6095537}a^{19}-\frac{2468645}{6095537}a^{18}+\frac{13423290}{6095537}a^{17}+\frac{24458515}{6095537}a^{16}+\frac{2470403}{6095537}a^{15}-\frac{61792193}{6095537}a^{14}+\frac{12012998}{6095537}a^{13}+\frac{135044531}{6095537}a^{12}+\frac{59344066}{6095537}a^{11}-\frac{88736094}{6095537}a^{10}-\frac{135617077}{6095537}a^{9}+\frac{95168233}{6095537}a^{8}+\frac{247718587}{6095537}a^{7}+\frac{42957654}{6095537}a^{6}-\frac{97123202}{6095537}a^{5}-\frac{112422672}{6095537}a^{4}-\frac{7998726}{6095537}a^{3}+\frac{71192463}{6095537}a^{2}+\frac{6805847}{6095537}a-\frac{4177671}{6095537}$, $\frac{3828322}{9578701}a^{20}-\frac{783098}{67050907}a^{19}-\frac{50059839}{67050907}a^{18}+\frac{31748593}{67050907}a^{17}+\frac{172749974}{67050907}a^{16}+\frac{27007069}{9578701}a^{15}-\frac{194958361}{67050907}a^{14}-\frac{39187079}{9578701}a^{13}+\frac{77224518}{9578701}a^{12}+\frac{1027922908}{67050907}a^{11}+\frac{37166927}{9578701}a^{10}-\frac{846234808}{67050907}a^{9}-\frac{442419401}{67050907}a^{8}+\frac{1290723344}{67050907}a^{7}+\frac{1650909879}{67050907}a^{6}+\frac{414358501}{67050907}a^{5}-\frac{642490883}{67050907}a^{4}-\frac{102421676}{9578701}a^{3}-\frac{40422499}{67050907}a^{2}+\frac{199562404}{67050907}a+\frac{72863269}{67050907}$, $\frac{50258345}{67050907}a^{20}-\frac{52389220}{67050907}a^{19}-\frac{46321008}{67050907}a^{18}+\frac{102617090}{67050907}a^{17}+\frac{236851423}{67050907}a^{16}+\frac{97357357}{67050907}a^{15}-\frac{486961026}{67050907}a^{14}-\frac{81113764}{67050907}a^{13}+\frac{1111087753}{67050907}a^{12}+\frac{871424593}{67050907}a^{11}-\frac{499848119}{67050907}a^{10}-\frac{1294404238}{67050907}a^{9}+\frac{327945747}{67050907}a^{8}+\frac{2141519764}{67050907}a^{7}+\frac{1077104519}{67050907}a^{6}-\frac{460370726}{67050907}a^{5}-\frac{1098107610}{67050907}a^{4}-\frac{546448211}{67050907}a^{3}+\frac{329666790}{67050907}a^{2}+\frac{248339249}{67050907}a+\frac{59141074}{67050907}$, $\frac{2705595}{3944171}a^{20}-\frac{4028547}{3944171}a^{19}-\frac{520180}{3944171}a^{18}+\frac{6235961}{3944171}a^{17}+\frac{9402754}{3944171}a^{16}+\frac{638140}{3944171}a^{15}-\frac{24547823}{3944171}a^{14}+\frac{1492892}{563453}a^{13}+\frac{56010700}{3944171}a^{12}+\frac{17008528}{3944171}a^{11}-\frac{32749174}{3944171}a^{10}-\frac{38609041}{3944171}a^{9}+\frac{48598562}{3944171}a^{8}+\frac{89856903}{3944171}a^{7}+\frac{1156360}{563453}a^{6}-\frac{2858385}{563453}a^{5}-\frac{21965931}{3944171}a^{4}+\frac{149351}{3944171}a^{3}+\frac{23578965}{3944171}a^{2}-\frac{1889574}{3944171}a-\frac{4405369}{3944171}$ 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:  \( 32527.0761402 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 32527.0761402 \cdot 1}{2\cdot\sqrt{156106141952973043121291857}}\cr\approx \mathstrut & 0.249651232568 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1)
 
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^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1, 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^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1);
 
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^21 - x^20 - x^19 + 2*x^18 + 5*x^17 + 2*x^16 - 10*x^15 - 2*x^14 + 23*x^13 + 19*x^12 - 11*x^11 - 28*x^10 + 8*x^9 + 47*x^8 + 23*x^7 - 13*x^6 - 24*x^5 - 7*x^4 + 11*x^3 + 6*x^2 - 1);
 
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

$S_3\times D_7$ (as 21T8):

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 solvable group of order 84
The 15 conjugacy class representatives for $S_3\times D_7$
Character table for $S_3\times D_7$

Intermediate fields

3.1.23.1, 7.1.357911.1

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]
 

Sibling fields

Degree 42 siblings: 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 $21$ $21$ ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ ${\href{/padicField/7.2.0.1}{2} }^{10}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{10}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ R $21$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(71\) Copy content Toggle raw display 71.3.0.1$x^{3} + 4 x + 64$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.3.2$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$