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Properties

Label 21.3.335...703.1
Degree $21$
Signature $[3, 9]$
Discriminant $-3.352\times 10^{23}$
Root discriminant \(13.19\)
Ramified primes $3,2447$
Class number $1$
Class group trivial
Galois group $D_7\wr C_3$ (as 21T45)

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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 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*x - 1)
 
Copy content gp:K = bnfinit(y^21 - 3*y^20 + 6*y^19 - 14*y^18 + 24*y^17 - 36*y^16 + 48*y^15 - 54*y^14 + 51*y^13 - 21*y^12 + 9*y^11 + 36*y^10 - 41*y^9 + 63*y^8 - 30*y^7 + 34*y^6 - 6*y^5 + 7*y^3 - 3*y^2 + 3*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*x - 1)
 

\( x^{21} - 3 x^{20} + 6 x^{19} - 14 x^{18} + 24 x^{17} - 36 x^{16} + 48 x^{15} - 54 x^{14} + 51 x^{13} + \cdots - 1 \) 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:  $[3, 9]$
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:   \(-335194620603462704888703\) \(\medspace = -\,3^{28}\cdot 2447^{3}\) 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:  \(13.19\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}2447^{1/2}\approx 214.03197432877192$
Ramified primes:   \(3\), \(2447\) 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:  \(\Q(\sqrt{-2447}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3516438736799}a^{20}-\frac{1095594088396}{3516438736799}a^{19}+\frac{1520020383480}{3516438736799}a^{18}+\frac{1395587027859}{3516438736799}a^{17}-\frac{1507328928725}{3516438736799}a^{16}-\frac{15856624350}{3516438736799}a^{15}-\frac{171069338844}{3516438736799}a^{14}-\frac{300330354981}{3516438736799}a^{13}+\frac{80228664672}{3516438736799}a^{12}-\frac{1262297793803}{3516438736799}a^{11}+\frac{217282525223}{3516438736799}a^{10}+\frac{855643305839}{3516438736799}a^{9}+\frac{1660519854962}{3516438736799}a^{8}+\frac{903033293712}{3516438736799}a^{7}-\frac{1718326130527}{3516438736799}a^{6}+\frac{873539162271}{3516438736799}a^{5}+\frac{1757790930504}{3516438736799}a^{4}+\frac{1239293514157}{3516438736799}a^{3}-\frac{266771440810}{3516438736799}a^{2}-\frac{167181132962}{3516438736799}a+\frac{151613034275}{3516438736799}$ 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 $1$
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 $1$
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:   $\frac{414739216674}{3516438736799}a^{20}+\frac{1193751919262}{3516438736799}a^{19}-\frac{4576583471477}{3516438736799}a^{18}+\frac{7332354333653}{3516438736799}a^{17}-\frac{20194341693421}{3516438736799}a^{16}+\frac{35333135944373}{3516438736799}a^{15}-\frac{51850991228568}{3516438736799}a^{14}+\frac{68174776075344}{3516438736799}a^{13}-\frac{73566962694729}{3516438736799}a^{12}+\frac{72004196471756}{3516438736799}a^{11}-\frac{4205308596470}{3516438736799}a^{10}+\frac{7086955464572}{3516438736799}a^{9}+\frac{72297924689449}{3516438736799}a^{8}-\frac{51202827582050}{3516438736799}a^{7}+\frac{98824834120061}{3516438736799}a^{6}-\frac{12610530965056}{3516438736799}a^{5}+\frac{37196431188786}{3516438736799}a^{4}+\frac{4723959349031}{3516438736799}a^{3}-\frac{2139833571487}{3516438736799}a^{2}+\frac{13847956172018}{3516438736799}a-\frac{844415500404}{3516438736799}$, $\frac{739226843868}{3516438736799}a^{20}+\frac{2006932987810}{3516438736799}a^{19}-\frac{2437660913151}{3516438736799}a^{18}+\frac{5420061263557}{3516438736799}a^{17}-\frac{8286787609548}{3516438736799}a^{16}+\frac{6271365717496}{3516438736799}a^{15}-\frac{3596289107441}{3516438736799}a^{14}-\frac{3788267435349}{3516438736799}a^{13}+\frac{14851566249884}{3516438736799}a^{12}-\frac{35360114544355}{3516438736799}a^{11}+\frac{28820482760662}{3516438736799}a^{10}-\frac{19674414362063}{3516438736799}a^{9}+\frac{13199755195918}{3516438736799}a^{8}+\frac{13563096235188}{3516438736799}a^{7}-\frac{24873304732484}{3516438736799}a^{6}+\frac{30659776321647}{3516438736799}a^{5}-\frac{1705306140258}{3516438736799}a^{4}+\frac{14290451109104}{3516438736799}a^{3}+\frac{5493116551456}{3516438736799}a^{2}-\frac{2353624975666}{3516438736799}a+\frac{6776203161924}{3516438736799}$, $a$, $\frac{196797211032}{3516438736799}a^{20}-\frac{645631536496}{3516438736799}a^{19}+\frac{1498665885046}{3516438736799}a^{18}-\frac{3896234543762}{3516438736799}a^{17}+\frac{7027328142756}{3516438736799}a^{16}-\frac{11227920073503}{3516438736799}a^{15}+\frac{17336043536800}{3516438736799}a^{14}-\frac{21206012857743}{3516438736799}a^{13}+\frac{22970442557862}{3516438736799}a^{12}-\frac{18062686891578}{3516438736799}a^{11}+\frac{11401703913578}{3516438736799}a^{10}+\frac{2626777720628}{3516438736799}a^{9}-\frac{15070231847106}{3516438736799}a^{8}+\frac{17109734948419}{3516438736799}a^{7}-\frac{19685444195926}{3516438736799}a^{6}+\frac{15356710457014}{3516438736799}a^{5}-\frac{8287608808646}{3516438736799}a^{4}+\frac{991665408251}{3516438736799}a^{3}-\frac{108895478145}{3516438736799}a^{2}+\frac{1185397698958}{3516438736799}a+\frac{3480700344266}{3516438736799}$, $\frac{915295302379}{3516438736799}a^{20}+\frac{3530693683349}{3516438736799}a^{19}-\frac{6898275595582}{3516438736799}a^{18}+\frac{14267277217496}{3516438736799}a^{17}-\frac{26478274189794}{3516438736799}a^{16}+\frac{37801448807191}{3516438736799}a^{15}-\frac{47456020110302}{3516438736799}a^{14}+\frac{51180464621683}{3516438736799}a^{13}-\frac{43820404449595}{3516438736799}a^{12}+\frac{11149859401864}{3516438736799}a^{11}+\frac{15848923659553}{3516438736799}a^{10}-\frac{32649769767176}{3516438736799}a^{9}+\frac{51616518759614}{3516438736799}a^{8}-\frac{45514310541507}{3516438736799}a^{7}+\frac{27060359181825}{3516438736799}a^{6}-\frac{1433812988432}{3516438736799}a^{5}+\frac{12799510770069}{3516438736799}a^{4}+\frac{1764747933490}{3516438736799}a^{3}-\frac{5543875922047}{3516438736799}a^{2}+\frac{590357002479}{3516438736799}a+\frac{2364635495980}{3516438736799}$, $\frac{279541031090}{3516438736799}a^{20}-\frac{1007587538257}{3516438736799}a^{19}+\frac{2422887259799}{3516438736799}a^{18}-\frac{6055762454443}{3516438736799}a^{17}+\frac{11680813763400}{3516438736799}a^{16}-\frac{19296165999665}{3516438736799}a^{15}+\frac{29394742564198}{3516438736799}a^{14}-\frac{38506976884920}{3516438736799}a^{13}+\frac{42907813300921}{3516438736799}a^{12}-\frac{36805689866328}{3516438736799}a^{11}+\frac{25688527962932}{3516438736799}a^{10}+\frac{332347276965}{3516438736799}a^{9}-\frac{22716092668458}{3516438736799}a^{8}+\frac{41221253083774}{3516438736799}a^{7}-\frac{41427168811350}{3516438736799}a^{6}+\frac{39884044068564}{3516438736799}a^{5}-\frac{20167865893171}{3516438736799}a^{4}+\frac{7801678554844}{3516438736799}a^{3}+\frac{2141551365086}{3516438736799}a^{2}-\frac{3695167159969}{3516438736799}a+\frac{4906376548258}{3516438736799}$, $\frac{798371563233}{3516438736799}a^{20}-\frac{2586050174275}{3516438736799}a^{19}+\frac{5146115854294}{3516438736799}a^{18}-\frac{11442105573124}{3516438736799}a^{17}+\frac{19514538650820}{3516438736799}a^{16}-\frac{28045818077234}{3516438736799}a^{15}+\frac{35425376201696}{3516438736799}a^{14}-\frac{36017126013834}{3516438736799}a^{13}+\frac{28155675687308}{3516438736799}a^{12}+\frac{922213509013}{3516438736799}a^{11}-\frac{16379565784795}{3516438736799}a^{10}+\frac{44342795657528}{3516438736799}a^{9}-\frac{42717900581453}{3516438736799}a^{8}+\frac{39116634526919}{3516438736799}a^{7}-\frac{7943519224079}{3516438736799}a^{6}-\frac{4389677537257}{3516438736799}a^{5}+\frac{12230920190989}{3516438736799}a^{4}-\frac{14275026868480}{3516438736799}a^{3}+\frac{1270588493510}{3516438736799}a^{2}+\frac{921178937315}{3516438736799}a-\frac{3957910209667}{3516438736799}$, $\frac{1035865516525}{3516438736799}a^{20}-\frac{2436040291470}{3516438736799}a^{19}+\frac{4419257199699}{3516438736799}a^{18}-\frac{10890419476085}{3516438736799}a^{17}+\frac{15970243263724}{3516438736799}a^{16}-\frac{22651682356550}{3516438736799}a^{15}+\frac{27316810022714}{3516438736799}a^{14}-\frac{25307051438575}{3516438736799}a^{13}+\frac{18149796891377}{3516438736799}a^{12}+\frac{11785514306872}{3516438736799}a^{11}-\frac{6211174658982}{3516438736799}a^{10}+\frac{48808070248012}{3516438736799}a^{9}-\frac{17767559848283}{3516438736799}a^{8}+\frac{42859792690341}{3516438736799}a^{7}+\frac{14266837759183}{3516438736799}a^{6}+\frac{16999059017351}{3516438736799}a^{5}+\frac{24309385965274}{3516438736799}a^{4}+\frac{3271308256912}{3516438736799}a^{3}+\frac{9196544847260}{3516438736799}a^{2}+\frac{4130258916312}{3516438736799}a+\frac{1048822203920}{3516438736799}$, $\frac{766135471404}{3516438736799}a^{20}-\frac{1927946603221}{3516438736799}a^{19}+\frac{4383885170379}{3516438736799}a^{18}-\frac{11317992019885}{3516438736799}a^{17}+\frac{18433492152834}{3516438736799}a^{16}-\frac{30039184259359}{3516438736799}a^{15}+\frac{42588151935682}{3516438736799}a^{14}-\frac{50115579992927}{3516438736799}a^{13}+\frac{50739460607158}{3516438736799}a^{12}-\frac{28141150024286}{3516438736799}a^{11}+\frac{21629747758335}{3516438736799}a^{10}+\frac{38588605733724}{3516438736799}a^{9}-\frac{35956873329280}{3516438736799}a^{8}+\frac{72226705098067}{3516438736799}a^{7}-\frac{32154431592173}{3516438736799}a^{6}+\frac{50865319519308}{3516438736799}a^{5}+\frac{5515971839260}{3516438736799}a^{4}-\frac{340066684537}{3516438736799}a^{3}+\frac{9913604488116}{3516438736799}a^{2}-\frac{3099890244650}{3516438736799}a+\frac{7113186406355}{3516438736799}$, $\frac{934207316653}{3516438736799}a^{20}+\frac{2788477175501}{3516438736799}a^{19}-\frac{6254895348198}{3516438736799}a^{18}+\frac{14944145625491}{3516438736799}a^{17}-\frac{25477180575682}{3516438736799}a^{16}+\frac{40444617833952}{3516438736799}a^{15}-\frac{55769125813822}{3516438736799}a^{14}+\frac{64873797100143}{3516438736799}a^{13}-\frac{64013497928519}{3516438736799}a^{12}+\frac{34105229646819}{3516438736799}a^{11}-\frac{18905427762851}{3516438736799}a^{10}-\frac{41953519419196}{3516438736799}a^{9}+\frac{44332904688932}{3516438736799}a^{8}-\frac{74983449607455}{3516438736799}a^{7}+\frac{35156712596241}{3516438736799}a^{6}-\frac{45022940418614}{3516438736799}a^{5}+\frac{1994377245852}{3516438736799}a^{4}-\frac{2789116987481}{3516438736799}a^{3}-\frac{1937742095669}{3516438736799}a^{2}-\frac{1139380432953}{3516438736799}a-\frac{3106966348827}{3516438736799}$, $\frac{1357412446424}{3516438736799}a^{20}-\frac{3059235462074}{3516438736799}a^{19}+\frac{5203790289690}{3516438736799}a^{18}-\frac{12978410560280}{3516438736799}a^{17}+\frac{18414946765713}{3516438736799}a^{16}-\frac{25157358185538}{3516438736799}a^{15}+\frac{29202222663721}{3516438736799}a^{14}-\frac{25707549805294}{3516438736799}a^{13}+\frac{16601492184258}{3516438736799}a^{12}+\frac{20606189174740}{3516438736799}a^{11}-\frac{5526763151737}{3516438736799}a^{10}+\frac{55532772797938}{3516438736799}a^{9}-\frac{13465592858800}{3516438736799}a^{8}+\frac{48192618960017}{3516438736799}a^{7}+\frac{20901791479920}{3516438736799}a^{6}+\frac{25882269246352}{3516438736799}a^{5}+\frac{20033354808500}{3516438736799}a^{4}+\frac{2524010011682}{3516438736799}a^{3}+\frac{11584366604777}{3516438736799}a^{2}+\frac{3981177479354}{3516438736799}a+\frac{2572303297022}{3516438736799}$ Copy content Toggle raw display
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:  \( 1491.359478528498 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 1491.359478528498 \cdot 1}{2\cdot\sqrt{335194620603462704888703}}\cr\approx \mathstrut & 0.157257955532060 \end{aligned}\]

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 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*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 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*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 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*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 = polynomial_ring(QQ); K, a = number_field(x^21 - 3*x^20 + 6*x^19 - 14*x^18 + 24*x^17 - 36*x^16 + 48*x^15 - 54*x^14 + 51*x^13 - 21*x^12 + 9*x^11 + 36*x^10 - 41*x^9 + 63*x^8 - 30*x^7 + 34*x^6 - 6*x^5 + 7*x^3 - 3*x^2 + 3*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

$D_7\wr C_3$ (as 21T45):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 8232
The 55 conjugacy class representatives for $D_7\wr C_3$
Character table for $D_7\wr C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\)

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 28 siblings: 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 $21$ R ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ $21$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ $21$ ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ $21$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.7.0.1}{7} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(3\) Copy content Toggle raw display 3.7.3.28a2.1$x^{21} + 6 x^{16} + 9 x^{14} + 12 x^{11} + 36 x^{9} + 15 x^{7} + 8 x^{6} + 36 x^{4} + 30 x^{2} + 10$$3$$7$$28$$C_{21}$not computed
\(2447\) Copy content Toggle raw display $\Q_{2447}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2447}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
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 $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$

Spectrum of ring of integers

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