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Properties

Label 21.1.54083457956...1488.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{26}\cdot 3^{19}\cdot 7^{21}\cdot 13^{7}\cdot 19^{19}$
Root discriminant $1505.79$
Ramified primes $2, 3, 7, 13, 19$
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![-7883676593, 23796592883, -24705378292, 13069935536, -8618999431, 6132900389, -1200362772, 885880792, -263126318, 7001358, -10560508, 3879176, 65310, 85498, 28852, -4872, 2639, 287, -112, 56, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 56*x^19 - 112*x^18 + 287*x^17 + 2639*x^16 - 4872*x^15 + 28852*x^14 + 85498*x^13 + 65310*x^12 + 3879176*x^11 - 10560508*x^10 + 7001358*x^9 - 263126318*x^8 + 885880792*x^7 - 1200362772*x^6 + 6132900389*x^5 - 8618999431*x^4 + 13069935536*x^3 - 24705378292*x^2 + 23796592883*x - 7883676593)
 
gp: K = bnfinit(x^21 - 7*x^20 + 56*x^19 - 112*x^18 + 287*x^17 + 2639*x^16 - 4872*x^15 + 28852*x^14 + 85498*x^13 + 65310*x^12 + 3879176*x^11 - 10560508*x^10 + 7001358*x^9 - 263126318*x^8 + 885880792*x^7 - 1200362772*x^6 + 6132900389*x^5 - 8618999431*x^4 + 13069935536*x^3 - 24705378292*x^2 + 23796592883*x - 7883676593, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 56 x^{19} - 112 x^{18} + 287 x^{17} + 2639 x^{16} - 4872 x^{15} + 28852 x^{14} + 85498 x^{13} + 65310 x^{12} + 3879176 x^{11} - 10560508 x^{10} + 7001358 x^{9} - 263126318 x^{8} + 885880792 x^{7} - 1200362772 x^{6} + 6132900389 x^{5} - 8618999431 x^{4} + 13069935536 x^{3} - 24705378292 x^{2} + 23796592883 x - 7883676593 \)

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:  \(5408345795652875901935821111247961815379871355609667224052053311488=2^{26}\cdot 3^{19}\cdot 7^{21}\cdot 13^{7}\cdot 19^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1505.79$
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:  $2, 3, 7, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{2} - \frac{5}{12}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{5}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{4} a^{6} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{24} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{8} - \frac{1}{6} a^{6} + \frac{1}{8} a^{4} + \frac{1}{6} a^{2} - \frac{1}{8}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{9} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{2736} a^{14} + \frac{1}{456} a^{13} + \frac{29}{2736} a^{12} + \frac{1}{76} a^{11} + \frac{13}{912} a^{10} - \frac{1}{456} a^{9} - \frac{113}{2736} a^{8} + \frac{13}{228} a^{7} - \frac{31}{144} a^{6} + \frac{1}{8} a^{5} - \frac{1}{48} a^{4} + \frac{1}{6} a^{3} + \frac{31}{144} a^{2} + \frac{3}{8} a - \frac{25}{144}$, $\frac{1}{79344} a^{15} - \frac{13}{79344} a^{14} - \frac{1567}{79344} a^{13} - \frac{1313}{79344} a^{12} + \frac{469}{26448} a^{11} + \frac{169}{26448} a^{10} - \frac{341}{79344} a^{9} + \frac{3215}{79344} a^{8} - \frac{175}{4176} a^{7} - \frac{13}{144} a^{6} - \frac{133}{1392} a^{5} - \frac{199}{1392} a^{4} + \frac{871}{4176} a^{3} + \frac{1559}{4176} a^{2} - \frac{733}{4176} a - \frac{1}{144}$, $\frac{1}{158688} a^{16} + \frac{1}{39672} a^{14} + \frac{55}{4408} a^{13} - \frac{49}{4959} a^{12} - \frac{101}{13224} a^{11} - \frac{241}{19836} a^{10} + \frac{7}{228} a^{9} - \frac{3037}{79344} a^{8} + \frac{119}{6612} a^{7} + \frac{415}{2088} a^{6} + \frac{11}{696} a^{5} - \frac{25}{261} a^{4} - \frac{43}{696} a^{3} - \frac{91}{261} a^{2} - \frac{7}{174} a + \frac{143}{288}$, $\frac{1}{476064} a^{17} + \frac{1}{238032} a^{15} - \frac{41}{238032} a^{14} - \frac{25}{2736} a^{13} - \frac{3121}{238032} a^{12} - \frac{283}{238032} a^{11} + \frac{295}{79344} a^{10} - \frac{3175}{119016} a^{9} - \frac{149}{12528} a^{8} + \frac{923}{26448} a^{7} - \frac{2341}{12528} a^{6} - \frac{523}{12528} a^{5} + \frac{113}{4176} a^{4} + \frac{2893}{12528} a^{3} - \frac{3815}{12528} a^{2} + \frac{595}{8352} a + \frac{121}{432}$, $\frac{1}{476064} a^{18} - \frac{1}{476064} a^{16} + \frac{1}{238032} a^{15} - \frac{5}{39672} a^{14} + \frac{3785}{238032} a^{13} - \frac{2309}{119016} a^{12} + \frac{67}{79344} a^{11} + \frac{4003}{238032} a^{10} - \frac{971}{238032} a^{9} - \frac{1181}{79344} a^{8} - \frac{197}{8208} a^{7} - \frac{1115}{6264} a^{6} - \frac{5}{144} a^{5} - \frac{1225}{6264} a^{4} + \frac{2221}{12528} a^{3} - \frac{167}{928} a^{2} - \frac{151}{12528} a + \frac{115}{288}$, $\frac{1}{1781907552} a^{19} - \frac{143}{148492296} a^{18} + \frac{355}{890953776} a^{17} + \frac{2129}{1781907552} a^{16} + \frac{31}{37123074} a^{15} - \frac{92065}{890953776} a^{14} - \frac{74578}{55684611} a^{13} - \frac{59683}{296984592} a^{12} - \frac{6738275}{890953776} a^{11} - \frac{5645873}{890953776} a^{10} - \frac{540187}{148492296} a^{9} - \frac{2085707}{111369222} a^{8} + \frac{12215165}{222738444} a^{7} - \frac{337577}{5210256} a^{6} - \frac{683099}{2930769} a^{5} - \frac{5849285}{46892304} a^{4} + \frac{85399}{10420512} a^{3} + \frac{328781}{46892304} a^{2} - \frac{1205777}{15630768} a - \frac{302173}{1077984}$, $\frac{1}{50301876775890337219889245659187439843102138362605807063744} a^{20} + \frac{641180661346953405443436650501029664727732791455}{25150938387945168609944622829593719921551069181302903531872} a^{19} + \frac{80050689051821120071143419824909579933097764742683}{465758118295280900184159682029513331880575355209313028368} a^{18} + \frac{2129721484965284694305919081295755327821607426954755}{25150938387945168609944622829593719921551069181302903531872} a^{17} - \frac{20324654965263618270946478603889896982806098985238623}{16767292258630112406629748553062479947700712787535269021248} a^{16} - \frac{4548710301306230447373715613712913476602232094274801}{6287734596986292152486155707398429980387767295325725882968} a^{15} - \frac{125359685825684183449116497209109797104640531964690047}{3143867298493146076243077853699214990193883647662862941484} a^{14} - \frac{151737907418396465831441310707414400869795392240689053}{8030312384401394830761373828095057446216816469126086696} a^{13} - \frac{177551724812071971265930268194737529025606761967032938343}{25150938387945168609944622829593719921551069181302903531872} a^{12} - \frac{46511166170430562321920269898410016159494020947327990875}{12575469193972584304972311414796859960775534590651451765936} a^{11} + \frac{88087289685963127080672461909829815194733727663106457}{12045468576602092246142060742142586169325224703689130044} a^{10} - \frac{22503391035114420067878579912562287608278353619558765969}{661866799682767594998542706041939997935554452139550092944} a^{9} - \frac{106586337735378204756766792144249539748093160819939102765}{25150938387945168609944622829593719921551069181302903531872} a^{8} + \frac{67536116184976718687654873201575371961387729181353220377}{2095911532328764050828718569132809993462589098441908627656} a^{7} + \frac{22422043752219625559300099101081684026097323988213100353}{165466699920691898749635676510484999483888613034887523236} a^{6} + \frac{34234407792201530959457362755011675585218185288182989091}{330933399841383797499271353020969998967777226069775046472} a^{5} + \frac{145239830550453595605410465187736047564522364231440734309}{882489066243690126664723608055919997247405936186066790592} a^{4} + \frac{564011715937866862951891408125554900777446734468501570085}{1323733599365535189997085412083879995871108904279100185888} a^{3} + \frac{1756116693666282881323382317516463301384595602937592131}{34835094720145662894660142423259999891344971165239478576} a^{2} - \frac{596680952949133948083906251474665932859841524454278318951}{1323733599365535189997085412083879995871108904279100185888} a - \frac{42432090137115976006897798219153018248426364899012724927}{91291972370036909654971407729922758335938545122696564544}$

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

3.1.2964.1, 7.1.1807654339634125248.8

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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$$[\ ]_{7}^{3}$
2.14.20.10$x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.7.6.1$x^{7} - 19$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
19.14.13.2$x^{14} + 76$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$