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Properties

Label 21.3.647...536.1
Degree $21$
Signature $[3, 9]$
Discriminant $-6.478\times 10^{35}$
Root discriminant \(50.73\)
Ramified primes $2,79,159017$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_7\wr S_3$ (as 21T62)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64)
 
gp: K = bnfinit(y^21 - 6*y^20 + 22*y^19 - 44*y^18 + 46*y^17 - 18*y^16 - 9*y^15 - 94*y^14 + 67*y^13 + 50*y^12 + 44*y^11 - 96*y^10 + 288*y^9 + 690*y^8 + 639*y^7 - 246*y^6 - 1218*y^5 - 1692*y^4 - 1384*y^3 - 784*y^2 - 288*y - 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64)
 

\( x^{21} - 6 x^{20} + 22 x^{19} - 44 x^{18} + 46 x^{17} - 18 x^{16} - 9 x^{15} - 94 x^{14} + 67 x^{13} + \cdots - 64 \) 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:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-647754154086026198212292912701505536\) \(\medspace = -\,2^{23}\cdot 79^{7}\cdot 159017^{3}\) 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:  \(50.73\)
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:  $2^{2}79^{1/2}159017^{1/2}\approx 14177.358287071678$
Ramified primes:   \(2\), \(79\), \(159017\) 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:  $\Q(\sqrt{-25124686}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{3}{16}a^{8}+\frac{3}{16}a^{7}-\frac{5}{16}a^{6}-\frac{3}{16}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{15}+\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{3}{16}a^{9}+\frac{1}{8}a^{6}+\frac{3}{16}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{16}+\frac{1}{32}a^{12}+\frac{1}{16}a^{11}+\frac{1}{32}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{7}{16}a^{7}+\frac{11}{32}a^{6}+\frac{1}{8}a^{5}-\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}+\frac{1}{32}a^{13}-\frac{1}{16}a^{12}+\frac{1}{32}a^{11}-\frac{1}{8}a^{10}-\frac{3}{16}a^{8}-\frac{13}{32}a^{7}-\frac{1}{4}a^{6}+\frac{1}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1984}a^{18}+\frac{1}{1984}a^{17}-\frac{15}{992}a^{16}-\frac{3}{124}a^{15}+\frac{9}{1984}a^{14}+\frac{11}{1984}a^{13}+\frac{85}{1984}a^{12}+\frac{5}{1984}a^{11}+\frac{117}{992}a^{10}-\frac{131}{992}a^{9}+\frac{485}{1984}a^{8}-\frac{949}{1984}a^{7}-\frac{91}{496}a^{6}+\frac{189}{992}a^{5}-\frac{137}{496}a^{4}+\frac{9}{62}a^{3}-\frac{1}{2}a^{2}-\frac{7}{62}a+\frac{11}{31}$, $\frac{1}{7936}a^{19}-\frac{1}{256}a^{17}+\frac{53}{3968}a^{16}+\frac{181}{7936}a^{15}-\frac{123}{3968}a^{14}-\frac{211}{3968}a^{13}-\frac{113}{1984}a^{12}-\frac{887}{7936}a^{11}-\frac{7}{64}a^{10}-\frac{617}{7936}a^{9}-\frac{841}{3968}a^{8}-\frac{2639}{7936}a^{7}+\frac{1921}{3968}a^{6}-\frac{525}{3968}a^{5}+\frac{581}{1984}a^{4}+\frac{119}{992}a^{3}-\frac{45}{496}a^{2}-\frac{33}{248}a+\frac{51}{124}$, $\frac{1}{31744}a^{20}-\frac{1}{31744}a^{19}+\frac{1}{31744}a^{18}-\frac{327}{31744}a^{17}-\frac{389}{31744}a^{16}+\frac{21}{31744}a^{15}-\frac{55}{1984}a^{14}+\frac{409}{15872}a^{13}+\frac{797}{31744}a^{12}-\frac{317}{31744}a^{11}-\frac{1685}{31744}a^{10}+\frac{471}{31744}a^{9}+\frac{4643}{31744}a^{8}-\frac{11487}{31744}a^{7}-\frac{539}{7936}a^{6}-\frac{5161}{15872}a^{5}+\frac{3953}{7936}a^{4}-\frac{1785}{3968}a^{3}-\frac{269}{1984}a^{2}-\frac{89}{992}a-\frac{195}{496}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:  $11$
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:   $\frac{113}{1024}a^{20}-\frac{1217}{1024}a^{19}+\frac{6465}{1024}a^{18}-\frac{21943}{1024}a^{17}+\frac{49483}{1024}a^{16}-\frac{75659}{1024}a^{15}+\frac{609}{8}a^{14}-\frac{30367}{512}a^{13}+\frac{69789}{1024}a^{12}-\frac{80909}{1024}a^{11}+\frac{54043}{1024}a^{10}-\frac{36201}{1024}a^{9}+\frac{79139}{1024}a^{8}-\frac{126127}{1024}a^{7}-\frac{12403}{256}a^{6}-\frac{26137}{512}a^{5}+\frac{25761}{256}a^{4}+\frac{15655}{128}a^{3}+\frac{8467}{64}a^{2}+\frac{1719}{32}a+\frac{429}{16}$, $\frac{113}{1024}a^{20}-\frac{1217}{1024}a^{19}+\frac{6465}{1024}a^{18}-\frac{21943}{1024}a^{17}+\frac{49483}{1024}a^{16}-\frac{75659}{1024}a^{15}+\frac{609}{8}a^{14}-\frac{30367}{512}a^{13}+\frac{69789}{1024}a^{12}-\frac{80909}{1024}a^{11}+\frac{54043}{1024}a^{10}-\frac{36201}{1024}a^{9}+\frac{79139}{1024}a^{8}-\frac{126127}{1024}a^{7}-\frac{12403}{256}a^{6}-\frac{26137}{512}a^{5}+\frac{25761}{256}a^{4}+\frac{15655}{128}a^{3}+\frac{8467}{64}a^{2}+\frac{1719}{32}a+\frac{397}{16}$, $\frac{1397}{15872}a^{20}+\frac{6895}{15872}a^{19}-\frac{81795}{15872}a^{18}+\frac{413673}{15872}a^{17}-\frac{1154577}{15872}a^{16}+\frac{1926413}{15872}a^{15}-\frac{60801}{496}a^{14}+\frac{443189}{7936}a^{13}-\frac{1412775}{15872}a^{12}+\frac{2521115}{15872}a^{11}-\frac{960601}{15872}a^{10}-\frac{293681}{15872}a^{9}-\frac{429089}{15872}a^{8}+\frac{6633089}{15872}a^{7}+\frac{1237435}{3968}a^{6}-\frac{736617}{7936}a^{5}-\frac{2451135}{3968}a^{4}-\frac{1361385}{1984}a^{3}-\frac{479109}{992}a^{2}-\frac{94873}{496}a-\frac{11707}{248}$, $\frac{705}{1984}a^{20}-\frac{4727}{1984}a^{19}+\frac{1169}{124}a^{18}-\frac{695}{32}a^{17}+\frac{57733}{1984}a^{16}-\frac{38909}{1984}a^{15}-\frac{6165}{1984}a^{14}-\frac{27257}{1984}a^{13}+\frac{4653}{248}a^{12}+\frac{573}{31}a^{11}-\frac{30031}{1984}a^{10}-\frac{16689}{1984}a^{9}+\frac{3223}{32}a^{8}+\frac{90045}{496}a^{7}+\frac{34695}{496}a^{6}-\frac{67065}{496}a^{5}-\frac{81889}{248}a^{4}-\frac{21193}{62}a^{3}-\frac{7302}{31}a^{2}-\frac{3175}{31}a-\frac{923}{31}$, $\frac{122783}{31744}a^{20}-\frac{914815}{31744}a^{19}+\frac{3928399}{31744}a^{18}-\frac{10364937}{31744}a^{17}+\frac{17547077}{31744}a^{16}-\frac{19484245}{31744}a^{15}+\frac{423073}{992}a^{14}-\frac{8202449}{15872}a^{13}+\frac{22140275}{31744}a^{12}-\frac{13317203}{31744}a^{11}+\frac{7598613}{31744}a^{10}-\frac{13243575}{31744}a^{9}+\frac{49059245}{31744}a^{8}+\frac{23985839}{31744}a^{7}+\frac{1128475}{7936}a^{6}-\frac{29710727}{15872}a^{5}-\frac{17236225}{7936}a^{4}-\frac{7467287}{3968}a^{3}-\frac{1455875}{1984}a^{2}-\frac{279863}{992}a+\frac{14099}{496}$, $\frac{62611}{31744}a^{20}-\frac{565003}{31744}a^{19}+\frac{2697379}{31744}a^{18}-\frac{8140493}{31744}a^{17}+\frac{16001361}{31744}a^{16}-\frac{20746313}{31744}a^{15}+\frac{1079497}{1984}a^{14}-\frac{7058485}{15872}a^{13}+\frac{22305367}{31744}a^{12}-\frac{22073231}{31744}a^{11}+\frac{9856945}{31744}a^{10}-\frac{11266099}{31744}a^{9}+\frac{36885657}{31744}a^{8}-\frac{25871797}{31744}a^{7}-\frac{7802557}{7936}a^{6}-\frac{21681539}{15872}a^{5}+\frac{4171707}{7936}a^{4}+\frac{5538205}{3968}a^{3}+\frac{3570481}{1984}a^{2}+\frac{847373}{992}a+\frac{183967}{496}$, $\frac{4941}{31744}a^{20}+\frac{82019}{31744}a^{19}-\frac{734131}{31744}a^{18}+\frac{3445941}{31744}a^{17}-\frac{9620897}{31744}a^{16}+\frac{17108865}{31744}a^{15}-\frac{1261149}{1984}a^{14}+\frac{7516837}{15872}a^{13}-\frac{17343047}{31744}a^{12}+\frac{22660599}{31744}a^{11}-\frac{14467601}{31744}a^{10}+\frac{8391739}{31744}a^{9}-\frac{11113241}{31744}a^{8}+\frac{48188861}{31744}a^{7}+\frac{5295225}{7936}a^{6}+\frac{1742523}{15872}a^{5}-\frac{14523955}{7936}a^{4}-\frac{8387141}{3968}a^{3}-\frac{3687369}{1984}a^{2}-\frac{787045}{992}a-\frac{157127}{496}$, $\frac{96133}{31744}a^{20}-\frac{609301}{31744}a^{19}+\frac{2298357}{31744}a^{18}-\frac{4844451}{31744}a^{17}+\frac{5368391}{31744}a^{16}-\frac{1722727}{31744}a^{15}-\frac{109059}{992}a^{14}-\frac{1869371}{15872}a^{13}+\frac{3533537}{31744}a^{12}+\frac{9282959}{31744}a^{11}-\frac{4794537}{31744}a^{10}-\frac{4231773}{31744}a^{9}+\frac{25745087}{31744}a^{8}+\frac{62133701}{31744}a^{7}+\frac{7787513}{7936}a^{6}-\frac{20311917}{15872}a^{5}-\frac{27421819}{7936}a^{4}-\frac{14222605}{3968}a^{3}-\frac{4788913}{1984}a^{2}-\frac{990621}{992}a-\frac{131359}{496}$, $\frac{69763}{31744}a^{20}-\frac{510979}{31744}a^{19}+\frac{2178307}{31744}a^{18}-\frac{5702933}{31744}a^{17}+\frac{9659921}{31744}a^{16}-\frac{11044545}{31744}a^{15}+\frac{537035}{1984}a^{14}-\frac{5596821}{15872}a^{13}+\frac{13548791}{31744}a^{12}-\frac{7553847}{31744}a^{11}+\frac{5010273}{31744}a^{10}-\frac{7714427}{31744}a^{9}+\frac{26318697}{31744}a^{8}+\frac{17891171}{31744}a^{7}+\frac{1854279}{7936}a^{6}-\frac{16087611}{15872}a^{5}-\frac{11360701}{7936}a^{4}-\frac{5344491}{3968}a^{3}-\frac{1359287}{1984}a^{2}-\frac{267115}{992}a-\frac{14825}{496}$, $\frac{69}{7936}a^{20}+\frac{841}{256}a^{19}-\frac{192675}{7936}a^{18}+\frac{820961}{7936}a^{17}-\frac{2132721}{7936}a^{16}+\frac{3523365}{7936}a^{15}-\frac{236077}{496}a^{14}+\frac{1243261}{3968}a^{13}-\frac{3345015}{7936}a^{12}+\frac{4615571}{7936}a^{11}-\frac{2617417}{7936}a^{10}+\frac{1633815}{7936}a^{9}-\frac{2994177}{7936}a^{8}+\frac{10616777}{7936}a^{7}+\frac{1557179}{1984}a^{6}+\frac{873903}{3968}a^{5}-\frac{3264767}{1984}a^{4}-\frac{2044417}{992}a^{3}-\frac{930493}{496}a^{2}-\frac{201305}{248}a-\frac{40751}{124}$, $\frac{870583}{31744}a^{20}-\frac{6210007}{31744}a^{19}+\frac{25806455}{31744}a^{18}-\frac{64730465}{31744}a^{17}+\frac{101420877}{31744}a^{16}-\frac{99474141}{31744}a^{15}+\frac{3333143}{1984}a^{14}-\frac{43316289}{15872}a^{13}+\frac{119093787}{31744}a^{12}-\frac{41825083}{31744}a^{11}+\frac{18652573}{31744}a^{10}-\frac{69538799}{31744}a^{9}+\frac{311739781}{31744}a^{8}+\frac{285752887}{31744}a^{7}+\frac{21021435}{7936}a^{6}-\frac{201220687}{15872}a^{5}-\frac{155324649}{7936}a^{4}-\frac{72265295}{3968}a^{3}-\frac{636549}{64}a^{2}-\frac{3956639}{992}a-\frac{374261}{496}$ 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:  \( 182519624586 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 182519624586 \cdot 1}{2\cdot\sqrt{647754154086026198212292912701505536}}\cr\approx \mathstrut & 13.8446910894312 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64)
 
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 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64, 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 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64);
 
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 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64);
 
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

$D_7\wr S_3$ (as 21T62):

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 16464
The 65 conjugacy class representatives for $D_7\wr S_3$ are not computed
Character table for $D_7\wr S_3$ is not computed

Intermediate fields

3.3.316.1

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 28 sibling: data not computed
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 ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ $21$ $21$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.2.0.1}{2} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $21$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(79\) Copy content Toggle raw display 79.7.0.1$x^{7} + 4 x + 76$$1$$7$$0$$C_7$$[\ ]^{7}$
79.14.7.1$x^{14} + 972349822084 x^{2} - 1459497082948084$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(159017\) Copy content Toggle raw display $\Q_{159017}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$