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Properties

Label 21.13.304...464.1
Degree $21$
Signature $[13, 4]$
Discriminant $3.044\times 10^{40}$
Root discriminant \(84.68\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_7\wr C_3.C_2$ (as 21T162)

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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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*x - 1)
 
Copy content gp:K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 44*y^16 - 908*y^15 - 256*y^14 + 2701*y^13 + 781*y^12 - 5138*y^11 - 1458*y^10 + 6313*y^9 + 1901*y^8 - 4745*y^7 - 1695*y^6 + 1879*y^5 + 877*y^4 - 201*y^3 - 146*y^2 - 22*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*x - 1)
 

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 44 x^{16} - 908 x^{15} - 256 x^{14} + 2701 x^{13} + \cdots - 1 \) 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:  $21$
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:  $[13, 4]$
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:   \(30442588627548161668838591975407708094464\) \(\medspace = 2^{14}\cdot 37^{7}\cdot 113\cdot 8273\cdot 385537\cdot 54305277821449\) 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:  \(84.68\)
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:  $2^{2/3}37^{1/2}113^{1/2}8273^{1/2}385537^{1/2}54305277821449^{1/2}\approx 42718137712460.88$
Ramified primes:   \(2\), \(37\), \(113\), \(8273\), \(385537\), \(54305277821449\) 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:  $\Q(\sqrt{72418\!\cdots\!82669}$)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$ 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:  $C_{2}$, which has order $2$ (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:  $16$
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:   $\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-10a^{17}+\frac{17}{2}a^{16}+\frac{169}{2}a^{15}-\frac{125}{2}a^{14}-\frac{783}{2}a^{13}+\frac{527}{2}a^{12}+1087a^{11}-\frac{1393}{2}a^{10}-\frac{3745}{2}a^{9}+\frac{2287}{2}a^{8}+2013a^{7}-\frac{2125}{2}a^{6}-\frac{2621}{2}a^{5}+463a^{4}+480a^{3}-41a^{2}-66a-7$, $a$, $a+1$, $\frac{1}{4}a^{20}+\frac{5}{4}a^{19}-\frac{13}{2}a^{18}-26a^{17}+\frac{273}{4}a^{16}+\frac{907}{4}a^{15}-392a^{14}-\frac{4305}{4}a^{13}+\frac{5595}{4}a^{12}+\frac{12175}{4}a^{11}-\frac{13155}{4}a^{10}-5326a^{9}+\frac{20275}{4}a^{8}+\frac{11619}{2}a^{7}-4732a^{6}-\frac{15213}{4}a^{5}+\frac{4597}{2}a^{4}+\frac{2729}{2}a^{3}-371a^{2}-\frac{345}{2}a-\frac{49}{4}$, $a^{20}+\frac{1}{2}a^{19}-\frac{87}{4}a^{18}-\frac{51}{4}a^{17}+\frac{399}{2}a^{16}+\frac{497}{4}a^{15}-\frac{4051}{4}a^{14}-\frac{2465}{4}a^{13}+\frac{12641}{4}a^{12}+1736a^{11}-\frac{25511}{4}a^{10}-2979a^{9}+\frac{16779}{2}a^{8}+\frac{13307}{4}a^{7}-\frac{13595}{2}a^{6}-\frac{4881}{2}a^{5}+\frac{11831}{4}a^{4}+\frac{4365}{4}a^{3}-440a^{2}-\frac{707}{4}a-\frac{25}{2}$, $\frac{1}{4}a^{20}-\frac{5}{4}a^{19}-4a^{18}+\frac{49}{2}a^{17}+\frac{99}{4}a^{16}-\frac{819}{4}a^{15}-62a^{14}+\frac{3793}{4}a^{13}-\frac{193}{4}a^{12}-\frac{10613}{4}a^{11}+\frac{2879}{4}a^{10}+4597a^{9}-\frac{7649}{4}a^{8}-\frac{9717}{2}a^{7}+2359a^{6}+\frac{11809}{4}a^{5}-1356a^{4}-919a^{3}+264a^{2}+\frac{193}{2}a+\frac{23}{4}$, $\frac{3}{4}a^{20}-\frac{5}{4}a^{19}-\frac{59}{4}a^{18}+\frac{93}{4}a^{17}+\frac{491}{4}a^{16}-187a^{15}-\frac{2233}{4}a^{14}+\frac{1703}{2}a^{13}+1498a^{12}-\frac{9579}{4}a^{11}-\frac{4785}{2}a^{10}+\frac{8439}{2}a^{9}+\frac{8711}{4}a^{8}-\frac{17987}{4}a^{7}-\frac{2013}{2}a^{6}+\frac{10719}{4}a^{5}+\frac{811}{4}a^{4}-\frac{3065}{4}a^{3}-28a^{2}+\frac{277}{4}a+\frac{23}{4}$, $5a^{20}+a^{19}-\frac{213}{2}a^{18}-\frac{69}{2}a^{17}+\frac{3829}{4}a^{16}+\frac{1521}{4}a^{15}-4754a^{14}-\frac{3993}{2}a^{13}+\frac{57827}{4}a^{12}+\frac{23151}{4}a^{11}-\frac{113197}{4}a^{10}-\frac{40917}{4}a^{9}+\frac{144029}{4}a^{8}+\frac{48517}{4}a^{7}-\frac{56417}{2}a^{6}-\frac{19371}{2}a^{5}+11846a^{4}+\frac{9295}{2}a^{3}-\frac{6453}{4}a^{2}-\frac{3023}{4}a-64$, $\frac{29}{4}a^{20}+\frac{5}{2}a^{19}-\frac{623}{4}a^{18}-\frac{281}{4}a^{17}+\frac{5639}{4}a^{16}+\frac{1433}{2}a^{15}-7044a^{14}-\frac{14549}{4}a^{13}+\frac{43087}{2}a^{12}+\frac{41633}{4}a^{11}-\frac{169615}{4}a^{10}-\frac{73197}{4}a^{9}+54203a^{8}+21467a^{7}-\frac{170299}{4}a^{6}-16803a^{5}+\frac{71473}{4}a^{4}+\frac{31275}{4}a^{3}-\frac{4753}{2}a^{2}-\frac{2491}{2}a-114$, $\frac{3}{4}a^{20}-\frac{1}{2}a^{19}-\frac{31}{2}a^{18}+8a^{17}+136a^{16}-56a^{15}-660a^{14}+234a^{13}+\frac{3895}{2}a^{12}-\frac{1299}{2}a^{11}-\frac{7281}{2}a^{10}+1166a^{9}+\frac{8697}{2}a^{8}-1192a^{7}-3206a^{6}+553a^{5}+1330a^{4}-\frac{35}{2}a^{3}-\frac{887}{4}a^{2}-31a-\frac{3}{4}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{39}{4}a^{16}+\frac{33}{4}a^{15}+\frac{159}{2}a^{14}-\frac{117}{2}a^{13}-\frac{1397}{4}a^{12}+\frac{947}{4}a^{11}+\frac{3569}{4}a^{10}-\frac{2389}{4}a^{9}-\frac{5369}{4}a^{8}+\frac{3663}{4}a^{7}+1143a^{6}-739a^{5}-\frac{931}{2}a^{4}+230a^{3}+\frac{175}{4}a^{2}-\frac{55}{4}a-2$, $\frac{113}{4}a^{20}-\frac{13}{2}a^{19}-\frac{1179}{2}a^{18}+\frac{97}{2}a^{17}+5198a^{16}+88a^{15}-\frac{101141}{4}a^{14}-\frac{3437}{2}a^{13}+74894a^{12}+6126a^{11}-\frac{565985}{4}a^{10}-\frac{24197}{2}a^{9}+\frac{689259}{4}a^{8}+\frac{39765}{2}a^{7}-\frac{516267}{4}a^{6}-23862a^{5}+\frac{104719}{2}a^{4}+\frac{30945}{2}a^{3}-\frac{28109}{4}a^{2}-\frac{5791}{2}a-\frac{477}{2}$, $\frac{17}{4}a^{20}-\frac{9}{4}a^{19}-88a^{18}+34a^{17}+771a^{16}-\frac{897}{4}a^{15}-\frac{14899}{4}a^{14}+\frac{3631}{4}a^{13}+\frac{21823}{2}a^{12}-\frac{10213}{4}a^{11}-\frac{80757}{4}a^{10}+\frac{9405}{2}a^{9}+\frac{94853}{4}a^{8}-4717a^{7}-\frac{67407}{4}a^{6}+\frac{7351}{4}a^{5}+\frac{12831}{2}a^{4}+\frac{655}{2}a^{3}-\frac{3297}{4}a^{2}-193a-11$, $39a^{20}-\frac{79}{4}a^{19}-\frac{3211}{4}a^{18}+\frac{1133}{4}a^{17}+6985a^{16}-\frac{6853}{4}a^{15}-\frac{133913}{4}a^{14}+6162a^{13}+\frac{389015}{4}a^{12}-15317a^{11}-\frac{715091}{4}a^{10}+\frac{97089}{4}a^{9}+\frac{419957}{2}a^{8}-16312a^{7}-151134a^{6}-\frac{20103}{4}a^{5}+\frac{237557}{4}a^{4}+\frac{46153}{4}a^{3}-\frac{15799}{2}a^{2}-2734a-212$, $\frac{9}{4}a^{20}-\frac{1}{4}a^{19}-\frac{191}{4}a^{18}-a^{17}+429a^{16}+\frac{177}{4}a^{15}-2134a^{14}-\frac{1159}{4}a^{13}+\frac{25995}{4}a^{12}+\frac{3337}{4}a^{11}-12703a^{10}-1333a^{9}+\frac{64393}{4}a^{8}+1575a^{7}-\frac{50539}{4}a^{6}-\frac{6059}{4}a^{5}+\frac{21745}{4}a^{4}+922a^{3}-\frac{3421}{4}a^{2}-151a-4$, $\frac{37}{2}a^{20}-\frac{35}{4}a^{19}-\frac{765}{2}a^{18}+\frac{495}{4}a^{17}+3345a^{16}-740a^{15}-\frac{64519}{4}a^{14}+\frac{5371}{2}a^{13}+47218a^{12}-\frac{27963}{4}a^{11}-\frac{350501}{4}a^{10}+\frac{24021}{2}a^{9}+\frac{416045}{4}a^{8}-\frac{39009}{4}a^{7}-\frac{302483}{4}a^{6}+\frac{517}{4}a^{5}+30001a^{4}+\frac{16713}{4}a^{3}-4152a^{2}-\frac{4327}{4}a-\frac{279}{4}$ 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:  \( 5098826311570 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 5098826311570 \cdot 1}{2\cdot\sqrt{30442588627548161668838591975407708094464}}\cr\approx \mathstrut & 0.186555840528488 \end{aligned}\] (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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 44*x^16 - 908*x^15 - 256*x^14 + 2701*x^13 + 781*x^12 - 5138*x^11 - 1458*x^10 + 6313*x^9 + 1901*x^8 - 4745*x^7 - 1695*x^6 + 1879*x^5 + 877*x^4 - 201*x^3 - 146*x^2 - 22*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

$S_7\wr C_3.C_2$ (as 21T162):

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 768144384000
The 920 conjugacy class representatives for $S_7\wr C_3.C_2$
Character table for $S_7\wr C_3.C_2$

Intermediate fields

3.3.148.1

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 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 R ${\href{/padicField/3.9.0.1}{9} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $15{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ R ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ $18{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

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 2.7.3.14a1.1$x^{21} + 3 x^{15} + 3 x^{14} + 3 x^{9} + 6 x^{8} + 3 x^{7} + x^{3} + 3 x^{2} + 3 x + 3$$3$$7$$14$21T6$$[\ ]_{3}^{14}$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.6.1.0a1.1$x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
37.5.2.5a1.2$x^{10} + 20 x^{6} + 70 x^{5} + 100 x^{2} + 700 x + 1262$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(113\) Copy content Toggle raw display $\Q_{113}$$x + 110$$1$$1$$0$Trivial$$[\ ]$$
113.2.1.0a1.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.1.2.1a1.1$x^{2} + 113$$2$$1$$1$$C_2$$$[\ ]_{2}$$
113.4.1.0a1.1$x^{4} + 62 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
113.4.1.0a1.1$x^{4} + 62 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
113.8.1.0a1.1$x^{8} + 3 x^{4} + 98 x^{3} + 38 x^{2} + 28 x + 3$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(8273\) Copy content Toggle raw display $\Q_{8273}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{8273}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $10$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
\(385537\) Copy content Toggle raw display $\Q_{385537}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(54305277821449\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$

Spectrum of ring of integers

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