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Properties

Label 21.3.58922863816...1343.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 293^{18}$
Root discriminant $628.90$
Ramified primes $7, 293$
Class number $49$ (GRH)
Class group $[7, 7]$ (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-131863811763929, -12893672008602, 53152215171034, 529795117999, -7349761489371, 655623718393, 400745644202, -84195862092, -2778664855, -506512648, 81259502, 40651037, -7944166, -3960985, 25168, 76879, -17000, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 17000*x^16 + 76879*x^15 + 25168*x^14 - 3960985*x^13 - 7944166*x^12 + 40651037*x^11 + 81259502*x^10 - 506512648*x^9 - 2778664855*x^8 - 84195862092*x^7 + 400745644202*x^6 + 655623718393*x^5 - 7349761489371*x^4 + 529795117999*x^3 + 53152215171034*x^2 - 12893672008602*x - 131863811763929)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 17000*x^16 + 76879*x^15 + 25168*x^14 - 3960985*x^13 - 7944166*x^12 + 40651037*x^11 + 81259502*x^10 - 506512648*x^9 - 2778664855*x^8 - 84195862092*x^7 + 400745644202*x^6 + 655623718393*x^5 - 7349761489371*x^4 + 529795117999*x^3 + 53152215171034*x^2 - 12893672008602*x - 131863811763929, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 17000 x^{16} + 76879 x^{15} + 25168 x^{14} - 3960985 x^{13} - 7944166 x^{12} + 40651037 x^{11} + 81259502 x^{10} - 506512648 x^{9} - 2778664855 x^{8} - 84195862092 x^{7} + 400745644202 x^{6} + 655623718393 x^{5} - 7349761489371 x^{4} + 529795117999 x^{3} + 53152215171034 x^{2} - 12893672008602 x - 131863811763929 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-58922863816834248701730948403017453588761267745332101431343=-\,7^{17}\cdot 293^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $628.90$
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:  $7, 293$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} + \frac{1}{16} a^{14} + \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{5}{16} a^{6} + \frac{1}{16} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{7}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{16} a^{11} + \frac{3}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3} + \frac{3}{16} a^{2} + \frac{3}{8} a - \frac{7}{16}$, $\frac{1}{164080} a^{18} + \frac{73}{82040} a^{17} + \frac{199}{164080} a^{16} - \frac{10153}{164080} a^{15} - \frac{3561}{164080} a^{14} + \frac{3253}{164080} a^{13} + \frac{8103}{82040} a^{12} - \frac{137}{11720} a^{11} + \frac{2649}{23440} a^{10} + \frac{8927}{164080} a^{9} - \frac{40189}{164080} a^{8} + \frac{8003}{41020} a^{7} + \frac{44211}{164080} a^{6} + \frac{1407}{2930} a^{5} + \frac{3101}{23440} a^{4} - \frac{1489}{23440} a^{3} + \frac{1913}{11720} a^{2} - \frac{1539}{5860} a + \frac{11083}{23440}$, $\frac{1}{1148560} a^{19} - \frac{3}{1148560} a^{18} - \frac{4311}{229712} a^{17} + \frac{76}{71785} a^{16} + \frac{8129}{287140} a^{15} - \frac{20219}{574280} a^{14} + \frac{105789}{1148560} a^{13} + \frac{31657}{287140} a^{12} + \frac{405}{4688} a^{11} - \frac{3147}{14357} a^{10} - \frac{4163}{287140} a^{9} + \frac{277373}{1148560} a^{8} + \frac{237843}{1148560} a^{7} + \frac{54553}{1148560} a^{6} - \frac{9803}{164080} a^{5} - \frac{35459}{82040} a^{4} + \frac{8867}{164080} a^{3} + \frac{3321}{16408} a^{2} - \frac{79593}{164080} a + \frac{65613}{164080}$, $\frac{1}{712575130642421982334684345767317122891016241663274522504634512396391013351044461792782853765944531691580962838998612960} a^{20} + \frac{17219145231837787387573865011668851686163040951085866920761514579545888004131633997399242018610429042465444592381}{71257513064242198233468434576731712289101624166327452250463451239639101335104446179278285376594453169158096283899861296} a^{19} + \frac{298007233673497492806840392997771691736172827498464388028894962798487003217986909275795003294972170544089073886831}{142515026128484396466936869153463424578203248332654904500926902479278202670208892358556570753188906338316192567799722592} a^{18} + \frac{43004232603395739114958580251601480892896251198480717979692084174898486526059742577050875044605226994252343005384887}{2908469920989477478917078962315580093432719353727651112263814336311800054494059027725644301085487884455432501383667808} a^{17} - \frac{827779721629282405517688495553611950390707336274711383315315961147326921982497802980520391191534719521164055191109669}{35628756532121099116734217288365856144550812083163726125231725619819550667552223089639142688297226584579048141949930648} a^{16} + \frac{116932641608932309987450787051083391732288828182693436233818635444609414592164908645607581069305435004817657259332493}{3635587401236846848646348702894475116790899192159563890329767920389750068117573784657055376356859855569290626729584760} a^{15} + \frac{25391992711975404674012683027217261891688461971269673045714340406185734116481043782649427426640697618617221618754204231}{712575130642421982334684345767317122891016241663274522504634512396391013351044461792782853765944531691580962838998612960} a^{14} + \frac{3986620325427020078095384815103571094484076620040114677988330006836783661518532615108840865249314921537649545846726841}{101796447234631711762097763681045303270145177380467788929233501770913001907292065970397550537992075955940137548428373280} a^{13} - \frac{30018258536106805811202260050385297816437559155168808222949012707311387100664465961786654012895586972154186694813415461}{356287565321210991167342172883658561445508120831637261252317256198195506675522230896391426882972265845790481419499306480} a^{12} + \frac{43161524169295936901987228708324199839549508995166129687918706045023269272074794363924236360904052689652541410901027839}{356287565321210991167342172883658561445508120831637261252317256198195506675522230896391426882972265845790481419499306480} a^{11} + \frac{7494539367889772998673718040740032672951211223330464141240842887079187582016041488906945560744988474052838453131989893}{142515026128484396466936869153463424578203248332654904500926902479278202670208892358556570753188906338316192567799722592} a^{10} - \frac{24947119460239284724188083981188656060228125827558521463856954350287606100336728328248882412874910482516076347857456899}{142515026128484396466936869153463424578203248332654904500926902479278202670208892358556570753188906338316192567799722592} a^{9} - \frac{156264948935079007627289479445301134935197538172808871154765627848746369523910493053553471978666364696269668366960057499}{712575130642421982334684345767317122891016241663274522504634512396391013351044461792782853765944531691580962838998612960} a^{8} + \frac{167030267445883557799618594409941752460471748921720972403540837192271491898152010705544161607291118345603309302700774}{4453594566515137389591777161045732018068851510395465765653965702477443833444027886204892836037153323072381017743741331} a^{7} - \frac{52119515461962909989455551144471179939597077510049389915658927573957974817259044847036978791650712438905812884692705727}{178143782660605495583671086441829280722754060415818630626158628099097753337761115448195713441486132922895240709749653240} a^{6} + \frac{11186239570221284616690966260462713335175098258087298528185434234071994667588261543865992704923070095936519892894563571}{50898223617315855881048881840522651635072588690233894464616750885456500953646032985198775268996037977970068774214186640} a^{5} + \frac{32272710535844820841854817037406797324114721743887489004031484170414510875017968760619613308131068037387884683420815067}{101796447234631711762097763681045303270145177380467788929233501770913001907292065970397550537992075955940137548428373280} a^{4} - \frac{4874456363254476552096077287049195594558256473252734460245952414699821277750502771586913227169122512035773317089403565}{10179644723463171176209776368104530327014517738046778892923350177091300190729206597039755053799207595594013754842837328} a^{3} - \frac{37835598852889168884603542800232004886529682620659924999775353795375853496522266476701539386440491849255642549500450759}{101796447234631711762097763681045303270145177380467788929233501770913001907292065970397550537992075955940137548428373280} a^{2} + \frac{7881623220239330674892399826526155256187189420886439745941160220070185204650813533443010002614740174021849480174354959}{20359289446926342352419552736209060654029035476093557785846700354182600381458413194079510107598415191188027509685674656} a - \frac{43015546610468809864246302046202279184553796412656731042738990048570394843900176922926835388744524042897239348822686389}{101796447234631711762097763681045303270145177380467788929233501770913001907292065970397550537992075955940137548428373280}$

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

Class group and class number

$C_{7}\times C_{7}$, which has order $49$ (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:  $11$
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:  \( 77992079112627040000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), Deg 7

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
293Data not computed