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Properties

Label 20.4.293...400.1
Degree $20$
Signature $[4, 8]$
Discriminant $2.932\times 10^{22}$
Root discriminant \(13.28\)
Ramified primes $2,5,5783$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.C_2^4:S_5$ (as 20T964)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 1)
 
Copy content gp:K = bnfinit(y^20 - 2*y^19 - 2*y^18 + 8*y^17 + 7*y^16 - 32*y^15 - y^14 + 60*y^13 + 3*y^12 - 108*y^11 + 14*y^10 + 112*y^9 - 4*y^8 - 96*y^7 - 21*y^6 + 52*y^5 + 26*y^4 - 18*y^3 - 22*y^2 - 8*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 1)
 

\( x^{20} - 2 x^{19} - 2 x^{18} + 8 x^{17} + 7 x^{16} - 32 x^{15} - x^{14} + 60 x^{13} + 3 x^{12} + \cdots - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[4, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(29319238827689141862400\) \(\medspace = 2^{20}\cdot 5^{2}\cdot 5783^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(13.28\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{15/8}5^{1/2}5783^{1/2}\approx 623.724552629587$
Ramified primes:   \(2\), \(5\), \(5783\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{3}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}-\frac{1}{2}a^{8}-\frac{3}{8}a^{7}+\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{176}a^{18}+\frac{1}{44}a^{17}+\frac{5}{176}a^{16}+\frac{7}{88}a^{15}+\frac{3}{88}a^{14}+\frac{15}{88}a^{13}-\frac{1}{16}a^{12}+\frac{3}{44}a^{11}+\frac{43}{88}a^{10}+\frac{9}{22}a^{9}+\frac{5}{22}a^{8}-\frac{9}{44}a^{7}-\frac{5}{44}a^{6}-\frac{3}{16}a^{4}-\frac{7}{88}a^{3}-\frac{25}{176}a^{2}-\frac{9}{88}a+\frac{31}{176}$, $\frac{1}{3344}a^{19}+\frac{5}{304}a^{17}-\frac{9}{418}a^{16}-\frac{135}{1672}a^{15}+\frac{113}{1672}a^{14}+\frac{793}{3344}a^{13}+\frac{3}{836}a^{12}+\frac{67}{209}a^{11}+\frac{733}{1672}a^{10}-\frac{465}{1672}a^{9}+\frac{29}{88}a^{8}+\frac{777}{1672}a^{7}-\frac{565}{1672}a^{6}-\frac{121}{304}a^{5}+\frac{72}{209}a^{4}-\frac{805}{3344}a^{3}+\frac{459}{1672}a^{2}+\frac{389}{3344}a+\frac{499}{1672}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{6027}{1672}a^{19}-\frac{703}{88}a^{18}-\frac{4659}{836}a^{17}+\frac{50613}{1672}a^{16}+\frac{3950}{209}a^{15}-\frac{100761}{836}a^{14}+\frac{36935}{1672}a^{13}+\frac{360535}{1672}a^{12}-\frac{61509}{1672}a^{11}-\frac{326319}{836}a^{10}+\frac{232165}{1672}a^{9}+\frac{4274}{11}a^{8}-\frac{178293}{1672}a^{7}-\frac{71321}{209}a^{6}+\frac{689}{76}a^{5}+\frac{333479}{1672}a^{4}+\frac{6777}{152}a^{3}-\frac{140367}{1672}a^{2}-\frac{12773}{209}a-\frac{19967}{1672}$, $\frac{1517}{836}a^{19}-\frac{361}{88}a^{18}-\frac{2179}{836}a^{17}+\frac{25645}{1672}a^{16}+\frac{3711}{418}a^{15}-\frac{51009}{836}a^{14}+\frac{11415}{836}a^{13}+\frac{180409}{1672}a^{12}-\frac{18587}{836}a^{11}-\frac{163941}{836}a^{10}+\frac{31795}{418}a^{9}+\frac{4259}{22}a^{8}-\frac{48229}{836}a^{7}-\frac{71873}{418}a^{6}+\frac{477}{76}a^{5}+\frac{170187}{1672}a^{4}+\frac{1667}{76}a^{3}-\frac{72977}{1672}a^{2}-\frac{26361}{836}a-\frac{12049}{1672}$, $\frac{714}{209}a^{19}-\frac{1457}{176}a^{18}-\frac{725}{209}a^{17}+\frac{97851}{3344}a^{16}+\frac{19161}{1672}a^{15}-\frac{193273}{1672}a^{14}+\frac{76491}{1672}a^{13}+\frac{639733}{3344}a^{12}-\frac{63203}{836}a^{11}-\frac{288687}{836}a^{10}+\frac{168517}{836}a^{9}+\frac{27585}{88}a^{8}-\frac{136349}{836}a^{7}-\frac{454265}{1672}a^{6}+\frac{4271}{76}a^{5}+\frac{557121}{3344}a^{4}+\frac{17643}{1672}a^{3}-\frac{247141}{3344}a^{2}-\frac{69907}{1672}a-\frac{21135}{3344}$, $\frac{13301}{1672}a^{19}-\frac{3545}{176}a^{18}-\frac{4523}{836}a^{17}+\frac{223695}{3344}a^{16}+\frac{34063}{1672}a^{15}-\frac{445571}{1672}a^{14}+\frac{110701}{836}a^{13}+\frac{1376137}{3344}a^{12}-\frac{326317}{1672}a^{11}-\frac{317931}{418}a^{10}+\frac{858345}{1672}a^{9}+\frac{55309}{88}a^{8}-\frac{614825}{1672}a^{7}-\frac{955291}{1672}a^{6}+\frac{2538}{19}a^{5}+\frac{1159157}{3344}a^{4}+\frac{21191}{836}a^{3}-\frac{523761}{3344}a^{2}-\frac{153017}{1672}a-\frac{47803}{3344}$, $\frac{64}{209}a^{19}-\frac{27}{44}a^{18}-\frac{128}{209}a^{17}+\frac{4195}{1672}a^{16}+\frac{466}{209}a^{15}-\frac{8549}{836}a^{14}-\frac{273}{836}a^{13}+\frac{16657}{836}a^{12}+\frac{189}{209}a^{11}-\frac{63083}{1672}a^{10}+\frac{2313}{418}a^{9}+\frac{3677}{88}a^{8}-\frac{1063}{418}a^{7}-\frac{67517}{1672}a^{6}-\frac{87}{19}a^{5}+\frac{39945}{1672}a^{4}+\frac{3379}{418}a^{3}-\frac{2048}{209}a^{2}-\frac{8849}{836}a-\frac{2993}{1672}$, $\frac{19967}{1672}a^{19}-\frac{2419}{88}a^{18}-\frac{26577}{1672}a^{17}+\frac{84527}{836}a^{16}+\frac{22289}{418}a^{15}-\frac{83818}{209}a^{14}+\frac{16505}{152}a^{13}+\frac{1161085}{1672}a^{12}-\frac{150317}{836}a^{11}-\frac{2094927}{1672}a^{10}+\frac{116522}{209}a^{9}+\frac{105481}{88}a^{8}-\frac{182379}{418}a^{7}-\frac{158049}{152}a^{6}+\frac{13751}{152}a^{5}+\frac{511563}{836}a^{4}+\frac{185663}{1672}a^{3}-\frac{433953}{1672}a^{2}-\frac{298907}{1672}a-\frac{654}{19}$, $\frac{3451}{304}a^{19}-\frac{5155}{176}a^{18}-\frac{19397}{3344}a^{17}+\frac{315453}{3344}a^{16}+\frac{5154}{209}a^{15}-\frac{315751}{836}a^{14}+\frac{693619}{3344}a^{13}+\frac{170745}{304}a^{12}-\frac{122437}{418}a^{11}-\frac{1762375}{1672}a^{10}+\frac{1285435}{1672}a^{9}+\frac{36285}{44}a^{8}-\frac{877551}{1672}a^{7}-\frac{649831}{836}a^{6}+\frac{63087}{304}a^{5}+\frac{142397}{304}a^{4}+\frac{85165}{3344}a^{3}-\frac{726129}{3344}a^{2}-\frac{409193}{3344}a-\frac{61323}{3344}$, $\frac{10819}{1672}a^{19}-\frac{1493}{88}a^{18}-\frac{4557}{1672}a^{17}+\frac{90757}{1672}a^{16}+\frac{2485}{209}a^{15}-\frac{181553}{836}a^{14}+\frac{19347}{152}a^{13}+\frac{535079}{1672}a^{12}-\frac{153203}{836}a^{11}-\frac{503089}{836}a^{10}+\frac{196025}{418}a^{9}+\frac{20397}{44}a^{8}-\frac{137321}{418}a^{7}-\frac{16751}{38}a^{6}+\frac{22003}{152}a^{5}+\frac{442435}{1672}a^{4}+\frac{7715}{1672}a^{3}-\frac{213189}{1672}a^{2}-\frac{113301}{1672}a-\frac{1435}{152}$, $\frac{28669}{3344}a^{19}-\frac{1945}{88}a^{18}-\frac{15063}{3344}a^{17}+\frac{14962}{209}a^{16}+\frac{30993}{1672}a^{15}-\frac{478285}{1672}a^{14}+\frac{524277}{3344}a^{13}+\frac{715623}{1672}a^{12}-\frac{376721}{1672}a^{11}-\frac{60801}{76}a^{10}+\frac{244791}{418}a^{9}+\frac{13899}{22}a^{8}-\frac{341371}{836}a^{7}-\frac{123544}{209}a^{6}+\frac{49281}{304}a^{5}+\frac{74906}{209}a^{4}+\frac{59199}{3344}a^{3}-\frac{141021}{836}a^{2}-\frac{314449}{3344}a-\frac{22867}{1672}$, $\frac{1809}{418}a^{19}-\frac{1565}{176}a^{18}-\frac{14909}{1672}a^{17}+\frac{123951}{3344}a^{16}+\frac{47265}{1672}a^{15}-\frac{243745}{1672}a^{14}+\frac{4841}{1672}a^{13}+\frac{945501}{3344}a^{12}-\frac{2999}{152}a^{11}-\frac{103582}{209}a^{10}+\frac{185323}{1672}a^{9}+\frac{4291}{8}a^{8}-\frac{178379}{1672}a^{7}-\frac{738427}{1672}a^{6}-\frac{4007}{152}a^{5}+\frac{875733}{3344}a^{4}+\frac{123559}{1672}a^{3}-\frac{337153}{3344}a^{2}-\frac{69031}{836}a-\frac{58295}{3344}$, $\frac{338}{209}a^{19}-\frac{257}{88}a^{18}-\frac{885}{209}a^{17}+\frac{5573}{418}a^{16}+\frac{11673}{836}a^{15}-\frac{11031}{209}a^{14}-\frac{5185}{418}a^{13}+\frac{183973}{1672}a^{12}+\frac{3369}{209}a^{11}-\frac{324255}{1672}a^{10}-\frac{270}{209}a^{9}+\frac{19887}{88}a^{8}+\frac{127}{418}a^{7}-\frac{318399}{1672}a^{6}-\frac{1749}{38}a^{5}+\frac{47879}{418}a^{4}+\frac{39309}{836}a^{3}-\frac{71271}{1672}a^{2}-\frac{8795}{209}a-\frac{8501}{836}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1357.71487099 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1357.71487099 \cdot 1}{2\cdot\sqrt{29319238827689141862400}}\cr\approx \mathstrut & 0.154085283980 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 2*x^18 + 8*x^17 + 7*x^16 - 32*x^15 - x^14 + 60*x^13 + 3*x^12 - 108*x^11 + 14*x^10 + 112*x^9 - 4*x^8 - 96*x^7 - 21*x^6 + 52*x^5 + 26*x^4 - 18*x^3 - 22*x^2 - 8*x - 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

$C_2^9.C_2^4:S_5$ (as 20T964):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 983040
The 155 conjugacy class representatives for $C_2^9.C_2^4:S_5$
Character table for $C_2^9.C_2^4:S_5$

Intermediate fields

5.3.5783.1, 10.6.34245723136.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.0.715801729191629440000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }^{4}$ R $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ $16{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.5.2.10a5.1$x^{10} + 2 x^{8} + 2 x^{7} + 4 x^{5} + x^{4} + 2 x^{3} + 2 x^{2} + 3$$2$$5$$10$$C_2^4 : C_5$$$[2, 2, 2, 2]^{5}$$
2.5.2.10a5.1$x^{10} + 2 x^{8} + 2 x^{7} + 4 x^{5} + x^{4} + 2 x^{3} + 2 x^{2} + 3$$2$$5$$10$$C_2^4 : C_5$$$[2, 2, 2, 2]^{5}$$
\(5\) Copy content Toggle raw display 5.3.1.0a1.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.3.1.0a1.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.6.1.0a1.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(5783\) Copy content Toggle raw display Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $8$$2$$4$$4$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)