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Properties

Label 21.21.1411787678...6672.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{15}\cdot 67^{7}\cdot 239^{3}\cdot 239339207^{3}$
Root discriminant $1016.56$
Ramified primes $2, 7, 67, 239, 239339207$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![851171, -1291883, -28197376, 64989214, 146533704, -379809836, -75283085, 413867107, -75688995, -162697023, 61218585, 23839329, -14368893, -549949, 1341897, -125375, -49095, 7773, 702, -154, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 154*x^19 + 702*x^18 + 7773*x^17 - 49095*x^16 - 125375*x^15 + 1341897*x^14 - 549949*x^13 - 14368893*x^12 + 23839329*x^11 + 61218585*x^10 - 162697023*x^9 - 75688995*x^8 + 413867107*x^7 - 75283085*x^6 - 379809836*x^5 + 146533704*x^4 + 64989214*x^3 - 28197376*x^2 - 1291883*x + 851171)
 
gp: K = bnfinit(x^21 - 3*x^20 - 154*x^19 + 702*x^18 + 7773*x^17 - 49095*x^16 - 125375*x^15 + 1341897*x^14 - 549949*x^13 - 14368893*x^12 + 23839329*x^11 + 61218585*x^10 - 162697023*x^9 - 75688995*x^8 + 413867107*x^7 - 75283085*x^6 - 379809836*x^5 + 146533704*x^4 + 64989214*x^3 - 28197376*x^2 - 1291883*x + 851171, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 154 x^{19} + 702 x^{18} + 7773 x^{17} - 49095 x^{16} - 125375 x^{15} + 1341897 x^{14} - 549949 x^{13} - 14368893 x^{12} + 23839329 x^{11} + 61218585 x^{10} - 162697023 x^{9} - 75688995 x^{8} + 413867107 x^{7} - 75283085 x^{6} - 379809836 x^{5} + 146533704 x^{4} + 64989214 x^{3} - 28197376 x^{2} - 1291883 x + 851171 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1411787678306418347597299085327799505357680312763424671062556672=2^{18}\cdot 7^{15}\cdot 67^{7}\cdot 239^{3}\cdot 239339207^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1016.56$
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:  $2, 7, 67, 239, 239339207$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{49} a^{11} + \frac{1}{49} a^{10} - \frac{1}{49} a^{9} + \frac{2}{49} a^{8} - \frac{3}{49} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{20}{49} a^{4} - \frac{15}{49} a^{3} + \frac{22}{49} a^{2} + \frac{12}{49} a + \frac{3}{49}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{2}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{7} a^{6} - \frac{22}{49} a^{5} + \frac{2}{7} a^{4} - \frac{12}{49} a^{3} - \frac{10}{49} a^{2} - \frac{16}{49} a + \frac{11}{49}$, $\frac{1}{147} a^{13} - \frac{1}{147} a^{12} - \frac{1}{21} a^{10} - \frac{10}{147} a^{9} - \frac{3}{49} a^{8} - \frac{3}{49} a^{7} + \frac{9}{49} a^{6} - \frac{9}{49} a^{5} + \frac{2}{21} a^{4} + \frac{5}{21} a^{3} + \frac{15}{49} a^{2} + \frac{65}{147} a + \frac{23}{147}$, $\frac{1}{147} a^{14} - \frac{1}{147} a^{12} - \frac{1}{147} a^{11} + \frac{10}{147} a^{10} - \frac{4}{147} a^{9} - \frac{2}{49} a^{8} - \frac{1}{7} a^{6} - \frac{13}{147} a^{5} + \frac{64}{147} a^{4} + \frac{11}{147} a^{3} - \frac{31}{147} a^{2} + \frac{55}{147} a - \frac{64}{147}$, $\frac{1}{147} a^{15} + \frac{1}{147} a^{12} + \frac{1}{147} a^{11} - \frac{5}{147} a^{10} + \frac{2}{147} a^{9} + \frac{2}{49} a^{7} + \frac{8}{21} a^{6} - \frac{29}{147} a^{5} - \frac{29}{147} a^{4} + \frac{19}{147} a^{3} - \frac{44}{147} a^{2} + \frac{55}{147} a - \frac{34}{147}$, $\frac{1}{6174} a^{16} + \frac{5}{3087} a^{15} - \frac{1}{2058} a^{14} - \frac{1}{882} a^{13} + \frac{1}{882} a^{12} - \frac{2}{441} a^{11} + \frac{13}{441} a^{10} - \frac{148}{3087} a^{9} + \frac{9}{343} a^{8} + \frac{178}{3087} a^{7} - \frac{45}{98} a^{6} + \frac{73}{441} a^{5} + \frac{149}{441} a^{4} - \frac{425}{882} a^{3} + \frac{1054}{3087} a^{2} - \frac{2377}{6174} a - \frac{1529}{6174}$, $\frac{1}{6174} a^{17} - \frac{19}{6174} a^{15} - \frac{19}{6174} a^{14} - \frac{1}{882} a^{13} - \frac{1}{441} a^{12} - \frac{1}{147} a^{11} - \frac{50}{3087} a^{10} + \frac{10}{441} a^{9} - \frac{2}{3087} a^{8} - \frac{95}{6174} a^{7} - \frac{185}{441} a^{6} + \frac{85}{441} a^{5} + \frac{23}{294} a^{4} - \frac{1543}{3087} a^{3} + \frac{47}{294} a^{2} + \frac{1199}{6174} a + \frac{988}{3087}$, $\frac{1}{43218} a^{18} - \frac{1}{14406} a^{16} + \frac{19}{14406} a^{15} - \frac{97}{43218} a^{14} + \frac{20}{3087} a^{12} - \frac{43}{21609} a^{11} - \frac{115}{3087} a^{10} + \frac{428}{7203} a^{9} - \frac{1913}{43218} a^{8} + \frac{104}{21609} a^{7} + \frac{184}{441} a^{6} - \frac{2623}{6174} a^{5} + \frac{9034}{21609} a^{4} - \frac{1121}{6174} a^{3} - \frac{8837}{43218} a^{2} + \frac{1565}{21609} a + \frac{5408}{21609}$, $\frac{1}{2593080} a^{19} + \frac{11}{2593080} a^{18} + \frac{11}{2593080} a^{17} + \frac{17}{2593080} a^{16} + \frac{649}{324135} a^{15} - \frac{389}{129654} a^{14} + \frac{139}{41160} a^{13} + \frac{1667}{518616} a^{12} - \frac{2794}{324135} a^{11} + \frac{13943}{324135} a^{10} + \frac{112393}{2593080} a^{9} - \frac{85529}{2593080} a^{8} + \frac{7367}{1296540} a^{7} + \frac{1373}{185220} a^{6} + \frac{1248269}{2593080} a^{5} + \frac{286843}{2593080} a^{4} - \frac{65431}{518616} a^{3} + \frac{1070107}{2593080} a^{2} - \frac{1231393}{2593080} a - \frac{1052761}{2593080}$, $\frac{1}{241241207877226634018550926300017457371200} a^{20} - \frac{10212218407335007919286078143372489}{60310301969306658504637731575004364342800} a^{19} + \frac{123789958652043289669745752487604119}{40206867979537772336425154383336242895200} a^{18} - \frac{23831110791562749618020431574450227}{574383828279111033377502205476232041360} a^{17} + \frac{667815928930333699367459691696256459}{8934859551008393852538923196296942865600} a^{16} - \frac{4444785260286787523583632566457051671}{1675286165814073847351048099305676787300} a^{15} - \frac{312242505681714319771819758796312397903}{241241207877226634018550926300017457371200} a^{14} + \frac{12701633742921567545180476266845055407}{30155150984653329252318865787502182171400} a^{13} - \frac{326664213014433079432406797987087416839}{80413735959075544672850308766672485790400} a^{12} - \frac{65306086581558183645907970048087895083}{10051716994884443084106288595834060723800} a^{11} - \frac{43472145184829184018520680942558458773}{765845104372148044503336273968309388480} a^{10} - \frac{7387359421151135123048848985704564699}{335057233162814769470209619861135357460} a^{9} + \frac{480365071369236049853348042154888324713}{26804578653025181557616769588890828596800} a^{8} - \frac{431812326919298175854939888537099448583}{20103433989768886168212577191668121447600} a^{7} + \frac{9566437530582998387027465611341528405419}{48248241575445326803710185260003491474240} a^{6} - \frac{242603301345427189018774013248972301261}{12062060393861331700927546315000872868560} a^{5} - \frac{2798518381606243172703427967052431867699}{10051716994884443084106288595834060723800} a^{4} - \frac{355591786622897060181896427415908184613}{717979785348888791721877756845290051700} a^{3} - \frac{4444577912099217217492527450377415700441}{120620603938613317009275463150008728685600} a^{2} + \frac{2320695187815248900866100684096377915319}{8041373595907554467285030876667248579040} a + \frac{11731845244821197378758242194085850170307}{241241207877226634018550926300017457371200}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 72248766090300000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 49392
The 51 conjugacy class representatives for t21n87 are not computed
Character table for t21n87 is not computed

Intermediate fields

3.3.469.1

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

Sibling fields

Degree 28 sibling: data not computed
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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ $21$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.24$x^{12} - 100 x^{10} - 59 x^{8} + 104 x^{6} + 387 x^{4} + 444 x^{2} + 439$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.0.1$x^{6} + x^{2} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
239Data not computed
239339207Data not computed