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Properties

Label 21.1.16856743381...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 5^{18}\cdot 7^{40}\cdot 97^{7}$
Root discriminant $2208.68$
Ramified primes $2, 5, 7, 97$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5460680552, -813508752, 3747573928, 1014152048, 954860816, 57561084, 118482154, 41542631, -4705764, 11794727, -3850, 3613071, -480760, 517055, -24570, 46669, -924, 2541, -14, 77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552)
 
gp: K = bnfinit(x^21 + 77*x^19 - 14*x^18 + 2541*x^17 - 924*x^16 + 46669*x^15 - 24570*x^14 + 517055*x^13 - 480760*x^12 + 3613071*x^11 - 3850*x^10 + 11794727*x^9 - 4705764*x^8 + 41542631*x^7 + 118482154*x^6 + 57561084*x^5 + 954860816*x^4 + 1014152048*x^3 + 3747573928*x^2 - 813508752*x + 5460680552, 1)
 

Normalized defining polynomial

\( x^{21} + 77 x^{19} - 14 x^{18} + 2541 x^{17} - 924 x^{16} + 46669 x^{15} - 24570 x^{14} + 517055 x^{13} - 480760 x^{12} + 3613071 x^{11} - 3850 x^{10} + 11794727 x^{9} - 4705764 x^{8} + 41542631 x^{7} + 118482154 x^{6} + 57561084 x^{5} + 954860816 x^{4} + 1014152048 x^{3} + 3747573928 x^{2} - 813508752 x + 5460680552 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16856743381788797964169462556045972312361821474422784000000000000000000=2^{33}\cdot 5^{18}\cdot 7^{40}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2208.68$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{392} a^{14} - \frac{1}{56} a^{13} + \frac{1}{56} a^{12} - \frac{1}{8} a^{11} + \frac{1}{56} a^{10} + \frac{5}{56} a^{9} - \frac{5}{56} a^{8} + \frac{41}{392} a^{7} - \frac{1}{14} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{2} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{2}{49}$, $\frac{1}{392} a^{15} - \frac{3}{28} a^{13} - \frac{3}{28} a^{11} - \frac{1}{28} a^{10} + \frac{1}{28} a^{9} - \frac{1}{49} a^{8} - \frac{5}{56} a^{7} + \frac{3}{28} a^{6} - \frac{1}{7} a^{5} - \frac{1}{2} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{49} a - \frac{2}{7}$, $\frac{1}{392} a^{16} - \frac{3}{28} a^{12} - \frac{1}{28} a^{11} + \frac{1}{28} a^{10} - \frac{1}{49} a^{9} - \frac{5}{56} a^{8} - \frac{1}{4} a^{7} + \frac{3}{28} a^{6} - \frac{1}{14} a^{4} - \frac{1}{7} a^{3} - \frac{2}{49} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{784} a^{17} - \frac{1}{784} a^{15} - \frac{1}{56} a^{12} + \frac{1}{14} a^{11} + \frac{3}{392} a^{10} - \frac{1}{16} a^{9} - \frac{45}{392} a^{8} - \frac{17}{112} a^{7} + \frac{11}{56} a^{6} - \frac{3}{14} a^{5} - \frac{1}{14} a^{4} - \frac{23}{98} a^{3} - \frac{1}{14} a^{2} + \frac{8}{49} a - \frac{5}{14}$, $\frac{1}{833392} a^{18} - \frac{171}{416696} a^{17} + \frac{443}{833392} a^{16} - \frac{73}{208348} a^{15} + \frac{185}{416696} a^{14} - \frac{538}{7441} a^{13} + \frac{2531}{59528} a^{12} - \frac{74}{52087} a^{11} + \frac{48901}{833392} a^{10} + \frac{57595}{416696} a^{9} - \frac{186013}{833392} a^{8} + \frac{19749}{208348} a^{7} - \frac{1901}{14882} a^{6} + \frac{6693}{14882} a^{5} - \frac{35583}{104174} a^{4} - \frac{7687}{52087} a^{3} + \frac{15097}{52087} a^{2} - \frac{50579}{104174} a - \frac{7615}{52087}$, $\frac{1}{11667488} a^{19} - \frac{1}{5833744} a^{18} + \frac{801}{1458436} a^{17} + \frac{377}{2916872} a^{16} - \frac{51}{11667488} a^{15} + \frac{4071}{5833744} a^{14} + \frac{46107}{416696} a^{13} + \frac{121771}{1458436} a^{12} - \frac{606653}{11667488} a^{11} - \frac{94967}{5833744} a^{10} + \frac{32883}{1458436} a^{9} + \frac{477859}{2916872} a^{8} + \frac{1071167}{11667488} a^{7} - \frac{138137}{833392} a^{6} + \frac{1078507}{2916872} a^{5} - \frac{267699}{729218} a^{4} - \frac{94055}{729218} a^{3} - \frac{97011}{364609} a^{2} + \frac{82384}{364609} a - \frac{353243}{1458436}$, $\frac{1}{6248830249469603954767227928089826125089298903236922735288195616} a^{20} + \frac{28218855382111951399270574690693486319453411999124526469}{3124415124734801977383613964044913062544649451618461367644097808} a^{19} - \frac{1252537990629639736322608100284024370954701127975439699511}{3124415124734801977383613964044913062544649451618461367644097808} a^{18} - \frac{65191777479006599854939611216447313850957318685112553368587}{111586254454814356335129070144461180805166051843516477415860636} a^{17} + \frac{828842088891918123231199923715658755070340175732452180641587}{6248830249469603954767227928089826125089298903236922735288195616} a^{16} - \frac{1968935190447451620781096926291281604558873023106484409912249}{3124415124734801977383613964044913062544649451618461367644097808} a^{15} - \frac{470515776228007176865527303570221770385717302309353110481049}{390551890591850247172951745505614132818081181452307670955512226} a^{14} + \frac{20273525641648883682556577920593785849321242643151248738633526}{195275945295925123586475872752807066409040590726153835477756113} a^{13} + \frac{359822733914011975593925767384341618759298385659230593658916983}{6248830249469603954767227928089826125089298903236922735288195616} a^{12} - \frac{249292012915593131043346932574901251971829191054805456073610317}{3124415124734801977383613964044913062544649451618461367644097808} a^{11} - \frac{4059243257590776986994159911847081938118083768033444523077889}{446345017819257425340516280577844723220664207374065909663442544} a^{10} + \frac{186657673545868811437190470214339402392793922014314305467999441}{781103781183700494345903491011228265636162362904615341911024452} a^{9} + \frac{363268472937205871415291848479667051631349329613895556955322949}{6248830249469603954767227928089826125089298903236922735288195616} a^{8} - \frac{687983754969667731015785959428753649863127837635090947933940023}{3124415124734801977383613964044913062544649451618461367644097808} a^{7} + \frac{324697498071816941184953213604934323774068618301069346608761695}{1562207562367400988691806982022456531272324725809230683822048904} a^{6} + \frac{201142254105641911360316186816393424879386254143883261791780855}{781103781183700494345903491011228265636162362904615341911024452} a^{5} - \frac{32074933189127991203267567674190184215070097029066442333286869}{195275945295925123586475872752807066409040590726153835477756113} a^{4} + \frac{10874662513149498513689067533349705521877672939849511254735401}{27896563613703589083782267536115295201291512960879119353965159} a^{3} + \frac{72081952045847459647653662582213704141006277896828347048800312}{195275945295925123586475872752807066409040590726153835477756113} a^{2} + \frac{351883557170705569050807741779281570693178623613298983380112895}{781103781183700494345903491011228265636162362904615341911024452} a + \frac{39973017475849739400485100496390918981070332092630866206930494}{195275945295925123586475872752807066409040590726153835477756113}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.5432.1, 7.1.96889010407000000.18

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

Sibling fields

Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.12.1$x^{14} - 5 x^{7} + 50$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.7.13.2$x^{7} + 105$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.55$x^{14} - 49 x^{13} - 147 x^{12} + 147 x^{11} - 49 x^{9} - 21 x^{7} - 98 x^{6} - 98 x^{5} + 147 x^{4} - 98 x^{3} - 49 x^{2} - 147 x + 140$$14$$1$$27$$F_7 \times C_2$$[13/6]_{6}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$