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Properties

Label 21.1.52000321857...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 5^{19}\cdot 7^{21}\cdot 211^{7}$
Root discriminant $323.81$
Ramified primes $2, 5, 7, 211$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![258383116, -678292132, 680126244, -99703156, -350328916, 268519692, -29714272, -56752004, 28662816, 2950976, -6850424, 1641416, 484596, -305914, 17861, 22673, -5313, -511, 329, -21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 21*x^19 + 329*x^18 - 511*x^17 - 5313*x^16 + 22673*x^15 + 17861*x^14 - 305914*x^13 + 484596*x^12 + 1641416*x^11 - 6850424*x^10 + 2950976*x^9 + 28662816*x^8 - 56752004*x^7 - 29714272*x^6 + 268519692*x^5 - 350328916*x^4 - 99703156*x^3 + 680126244*x^2 - 678292132*x + 258383116)
 
gp: K = bnfinit(x^21 - 7*x^20 - 21*x^19 + 329*x^18 - 511*x^17 - 5313*x^16 + 22673*x^15 + 17861*x^14 - 305914*x^13 + 484596*x^12 + 1641416*x^11 - 6850424*x^10 + 2950976*x^9 + 28662816*x^8 - 56752004*x^7 - 29714272*x^6 + 268519692*x^5 - 350328916*x^4 - 99703156*x^3 + 680126244*x^2 - 678292132*x + 258383116, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 21 x^{19} + 329 x^{18} - 511 x^{17} - 5313 x^{16} + 22673 x^{15} + 17861 x^{14} - 305914 x^{13} + 484596 x^{12} + 1641416 x^{11} - 6850424 x^{10} + 2950976 x^{9} + 28662816 x^{8} - 56752004 x^{7} - 29714272 x^{6} + 268519692 x^{5} - 350328916 x^{4} - 99703156 x^{3} + 680126244 x^{2} - 678292132 x + 258383116 \)

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:  \(52000321857243010504353469003759985000000000000000000=2^{18}\cdot 5^{19}\cdot 7^{21}\cdot 211^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $323.81$
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:  $2, 5, 7, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10} a^{14} - \frac{1}{2} a^{13} + \frac{3}{10} a^{12} - \frac{1}{2} a^{11} + \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{14} - \frac{7}{20} a^{13} + \frac{7}{20} a^{12} - \frac{9}{20} a^{11} + \frac{9}{20} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{20} a^{16} + \frac{1}{10} a^{12} - \frac{2}{5} a^{10} - \frac{1}{4} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{20} a^{17} + \frac{1}{10} a^{13} - \frac{2}{5} a^{11} - \frac{1}{4} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{18} - \frac{1}{2} a^{13} + \frac{3}{10} a^{12} - \frac{1}{2} a^{11} - \frac{7}{20} a^{10} - \frac{1}{2} a^{9} - \frac{3}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{100} a^{19} - \frac{1}{50} a^{17} + \frac{1}{100} a^{15} - \frac{1}{100} a^{14} + \frac{9}{20} a^{13} + \frac{7}{100} a^{12} - \frac{3}{10} a^{11} - \frac{11}{100} a^{10} + \frac{29}{100} a^{9} - \frac{3}{20} a^{8} - \frac{7}{25} a^{7} - \frac{1}{2} a^{6} + \frac{7}{50} a^{5} - \frac{1}{25} a^{4} - \frac{3}{10} a^{3} - \frac{1}{50} a^{2} + \frac{3}{10} a - \frac{1}{25}$, $\frac{1}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{20} + \frac{1223035375907997337037174361365001144401686783845551303531979631067629}{410567251065557796490982568265036592834750412986747255061207875600457575} a^{19} - \frac{37876138566246151744618910486383511362696279997610885277210456930718107}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{18} + \frac{8234632729740560847371224053678040769483073196296693284785182774250003}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{17} - \frac{38380377938239473935026816242466755986696024381946967018319589455383419}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{16} + \frac{392553147194133210084842851114349897634263970138005486070434774039867}{82113450213111559298196513653007318566950082597349451012241575120091515} a^{15} - \frac{1844662800157772252603084734896601184447821502207164102955124660664849}{410567251065557796490982568265036592834750412986747255061207875600457575} a^{14} - \frac{7689546430660760415897640624008654296317545441668242507551244565374407}{410567251065557796490982568265036592834750412986747255061207875600457575} a^{13} - \frac{781806416877354216156498479275157577188462570081519008856868989007610413}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{12} - \frac{20657705634346385110662503693855342230803954164953528808116848838871923}{821134502131115592981965136530073185669500825973494510122415751200915150} a^{11} - \frac{360172557400376446577332108811385303392765422428386641391220053820994927}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{10} - \frac{760502085301225659102946991359165542818707863440996147991514650477508101}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{9} + \frac{3091869099619276264182938946014349799010138826501701717911392862367287}{1642269004262231185963930273060146371339001651946989020244831502401830300} a^{8} + \frac{206450794965852064083438763103505715267908417748320862492680119352614221}{821134502131115592981965136530073185669500825973494510122415751200915150} a^{7} + \frac{7638026353527121343659501022023545021931326727011376740897882777913971}{410567251065557796490982568265036592834750412986747255061207875600457575} a^{6} + \frac{46055979520280174232051737460147197495971456200427144783924064797357317}{164226900426223118596393027306014637133900165194698902024483150240183030} a^{5} - \frac{171288737380075561955859158501684633217079446512640721833721795492718261}{410567251065557796490982568265036592834750412986747255061207875600457575} a^{4} + \frac{169090801162083180057316907924850596093426096073611180089051706451324749}{821134502131115592981965136530073185669500825973494510122415751200915150} a^{3} - \frac{113476897182987994689219285685809924776396207949484810899243159070979841}{821134502131115592981965136530073185669500825973494510122415751200915150} a^{2} - \frac{138722503062198384345238004351687192982331194380306003320631488738435547}{821134502131115592981965136530073185669500825973494510122415751200915150} a + \frac{187344658568539601604658029525755288117584752378183315504399054425118279}{410567251065557796490982568265036592834750412986747255061207875600457575}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  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:  \( 2795263802088440300 \) (assuming GRH)
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

3.1.1055.1, 7.1.823543000000.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.1$x^{14} - 5$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
211Data not computed