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Properties

Label 25.25.1982587808...6801.1
Degree $25$
Signature $[25, 0]$
Discriminant $701^{24}$
Root discriminant $539.37$
Ramified prime $701$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17666045410357, 8576334530877, -102219387211201, -51510706736207, 113659250436626, 67425341538954, -43871171250230, -29651886280369, 8667324085851, 6647166275045, -1019102628348, -885235325980, 76261714738, 74859514573, -3703105216, -4123860033, 113481379, 147831619, -1970280, -3362943, 13061, 45899, 75, -336, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 336*x^23 + 75*x^22 + 45899*x^21 + 13061*x^20 - 3362943*x^19 - 1970280*x^18 + 147831619*x^17 + 113481379*x^16 - 4123860033*x^15 - 3703105216*x^14 + 74859514573*x^13 + 76261714738*x^12 - 885235325980*x^11 - 1019102628348*x^10 + 6647166275045*x^9 + 8667324085851*x^8 - 29651886280369*x^7 - 43871171250230*x^6 + 67425341538954*x^5 + 113659250436626*x^4 - 51510706736207*x^3 - 102219387211201*x^2 + 8576334530877*x + 17666045410357)
 
gp: K = bnfinit(x^25 - x^24 - 336*x^23 + 75*x^22 + 45899*x^21 + 13061*x^20 - 3362943*x^19 - 1970280*x^18 + 147831619*x^17 + 113481379*x^16 - 4123860033*x^15 - 3703105216*x^14 + 74859514573*x^13 + 76261714738*x^12 - 885235325980*x^11 - 1019102628348*x^10 + 6647166275045*x^9 + 8667324085851*x^8 - 29651886280369*x^7 - 43871171250230*x^6 + 67425341538954*x^5 + 113659250436626*x^4 - 51510706736207*x^3 - 102219387211201*x^2 + 8576334530877*x + 17666045410357, 1)
 

Normalized defining polynomial

\( x^{25} - x^{24} - 336 x^{23} + 75 x^{22} + 45899 x^{21} + 13061 x^{20} - 3362943 x^{19} - 1970280 x^{18} + 147831619 x^{17} + 113481379 x^{16} - 4123860033 x^{15} - 3703105216 x^{14} + 74859514573 x^{13} + 76261714738 x^{12} - 885235325980 x^{11} - 1019102628348 x^{10} + 6647166275045 x^{9} + 8667324085851 x^{8} - 29651886280369 x^{7} - 43871171250230 x^{6} + 67425341538954 x^{5} + 113659250436626 x^{4} - 51510706736207 x^{3} - 102219387211201 x^{2} + 8576334530877 x + 17666045410357 \)

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

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(198258780827351695791008735816797746301096570719885588203276967256801=701^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $539.37$
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:  $701$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(701\)
Dirichlet character group:    $\lbrace$$\chi_{701}(320,·)$, $\chi_{701}(1,·)$, $\chi_{701}(387,·)$, $\chi_{701}(583,·)$, $\chi_{701}(456,·)$, $\chi_{701}(521,·)$, $\chi_{701}(13,·)$, $\chi_{701}(655,·)$, $\chi_{701}(464,·)$, $\chi_{701}(210,·)$, $\chi_{701}(600,·)$, $\chi_{701}(89,·)$, $\chi_{701}(154,·)$, $\chi_{701}(605,·)$, $\chi_{701}(94,·)$, $\chi_{701}(103,·)$, $\chi_{701}(424,·)$, $\chi_{701}(169,·)$, $\chi_{701}(112,·)$, $\chi_{701}(627,·)$, $\chi_{701}(54,·)$, $\chi_{701}(440,·)$, $\chi_{701}(569,·)$, $\chi_{701}(124,·)$, $\chi_{701}(638,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{8}{19} a^{9} + \frac{1}{19} a^{8} - \frac{7}{19} a^{7} + \frac{7}{19} a^{6} - \frac{2}{19} a^{5} - \frac{1}{19} a^{4} + \frac{2}{19} a^{3} + \frac{9}{19} a^{2} + \frac{1}{19} a$, $\frac{1}{19} a^{11} - \frac{6}{19} a^{9} + \frac{4}{19} a^{8} + \frac{6}{19} a^{7} - \frac{1}{19} a^{6} - \frac{4}{19} a^{5} - \frac{9}{19} a^{4} - \frac{7}{19} a^{3} + \frac{5}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{12} - \frac{5}{19} a^{9} - \frac{7}{19} a^{8} - \frac{5}{19} a^{7} - \frac{2}{19} a^{5} + \frac{6}{19} a^{4} - \frac{2}{19} a^{3} + \frac{8}{19} a^{2} + \frac{6}{19} a$, $\frac{1}{19} a^{13} - \frac{5}{19} a^{9} + \frac{3}{19} a^{7} - \frac{5}{19} a^{6} - \frac{4}{19} a^{5} - \frac{7}{19} a^{4} - \frac{1}{19} a^{3} - \frac{6}{19} a^{2} + \frac{5}{19} a$, $\frac{1}{19} a^{14} + \frac{2}{19} a^{9} + \frac{8}{19} a^{8} - \frac{2}{19} a^{7} - \frac{7}{19} a^{6} + \frac{2}{19} a^{5} - \frac{6}{19} a^{4} + \frac{4}{19} a^{3} - \frac{7}{19} a^{2} + \frac{5}{19} a$, $\frac{1}{19} a^{15} - \frac{8}{19} a^{9} - \frac{4}{19} a^{8} + \frac{7}{19} a^{7} + \frac{7}{19} a^{6} - \frac{2}{19} a^{5} + \frac{6}{19} a^{4} + \frac{8}{19} a^{3} + \frac{6}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{361} a^{16} - \frac{8}{361} a^{15} - \frac{5}{361} a^{14} + \frac{2}{361} a^{13} - \frac{9}{361} a^{11} - \frac{4}{361} a^{10} + \frac{107}{361} a^{9} - \frac{52}{361} a^{8} + \frac{75}{361} a^{7} + \frac{137}{361} a^{6} + \frac{70}{361} a^{5} + \frac{15}{361} a^{4} + \frac{143}{361} a^{3} + \frac{59}{361} a^{2} - \frac{151}{361} a - \frac{1}{19}$, $\frac{1}{361} a^{17} + \frac{7}{361} a^{15} - \frac{3}{361} a^{13} - \frac{9}{361} a^{12} - \frac{1}{361} a^{10} + \frac{25}{361} a^{9} - \frac{113}{361} a^{8} - \frac{42}{361} a^{7} - \frac{164}{361} a^{6} + \frac{62}{361} a^{5} + \frac{16}{361} a^{4} - \frac{146}{361} a^{3} - \frac{40}{361} a^{2} - \frac{163}{361} a - \frac{8}{19}$, $\frac{1}{361} a^{18} - \frac{1}{361} a^{15} - \frac{6}{361} a^{14} - \frac{4}{361} a^{13} + \frac{5}{361} a^{11} - \frac{4}{361} a^{10} + \frac{31}{361} a^{9} - \frac{39}{361} a^{8} - \frac{176}{361} a^{7} - \frac{23}{361} a^{6} - \frac{170}{361} a^{5} + \frac{72}{361} a^{4} + \frac{61}{361} a^{3} - \frac{120}{361} a^{2} - \frac{121}{361} a + \frac{7}{19}$, $\frac{1}{361} a^{19} + \frac{5}{361} a^{15} - \frac{9}{361} a^{14} + \frac{2}{361} a^{13} + \frac{5}{361} a^{12} + \frac{6}{361} a^{11} + \frac{8}{361} a^{10} + \frac{11}{361} a^{9} + \frac{6}{19} a^{8} + \frac{71}{361} a^{7} - \frac{52}{361} a^{6} + \frac{66}{361} a^{5} + \frac{2}{19} a^{4} + \frac{4}{361} a^{3} - \frac{24}{361} a^{2} + \frac{134}{361} a - \frac{1}{19}$, $\frac{1}{361} a^{20} - \frac{7}{361} a^{15} + \frac{8}{361} a^{14} - \frac{5}{361} a^{13} + \frac{6}{361} a^{12} - \frac{4}{361} a^{11} - \frac{7}{361} a^{10} - \frac{117}{361} a^{9} + \frac{65}{361} a^{8} - \frac{9}{361} a^{7} + \frac{122}{361} a^{6} + \frac{30}{361} a^{5} + \frac{5}{361} a^{4} - \frac{74}{361} a^{3} - \frac{161}{361} a^{2} + \frac{52}{361} a + \frac{5}{19}$, $\frac{1}{6859} a^{21} - \frac{4}{6859} a^{20} - \frac{6}{6859} a^{19} - \frac{8}{6859} a^{18} - \frac{1}{6859} a^{17} + \frac{6}{6859} a^{16} + \frac{93}{6859} a^{15} + \frac{7}{361} a^{14} + \frac{132}{6859} a^{13} - \frac{30}{6859} a^{12} + \frac{6}{6859} a^{11} + \frac{129}{6859} a^{10} - \frac{714}{6859} a^{9} - \frac{2515}{6859} a^{8} + \frac{2081}{6859} a^{7} + \frac{1736}{6859} a^{6} + \frac{709}{6859} a^{5} + \frac{3119}{6859} a^{4} - \frac{3312}{6859} a^{3} + \frac{2892}{6859} a^{2} + \frac{949}{6859} a - \frac{151}{361}$, $\frac{1}{6859} a^{22} - \frac{3}{6859} a^{20} + \frac{6}{6859} a^{19} + \frac{5}{6859} a^{18} + \frac{2}{6859} a^{17} + \frac{3}{6859} a^{16} - \frac{8}{6859} a^{15} + \frac{94}{6859} a^{14} + \frac{99}{6859} a^{13} - \frac{9}{361} a^{12} + \frac{77}{6859} a^{11} - \frac{84}{6859} a^{10} - \frac{507}{6859} a^{9} - \frac{1576}{6859} a^{8} - \frac{1929}{6859} a^{7} - \frac{916}{6859} a^{6} + \frac{3257}{6859} a^{5} - \frac{1989}{6859} a^{4} - \frac{2490}{6859} a^{3} + \frac{870}{6859} a^{2} + \frac{490}{6859} a - \frac{167}{361}$, $\frac{1}{28107571549} a^{23} - \frac{1001164}{28107571549} a^{22} + \frac{1466730}{28107571549} a^{21} + \frac{27277026}{28107571549} a^{20} - \frac{34065549}{28107571549} a^{19} + \frac{10837697}{28107571549} a^{18} - \frac{639604}{1479345871} a^{17} + \frac{14808370}{28107571549} a^{16} + \frac{556962366}{28107571549} a^{15} - \frac{545518461}{28107571549} a^{14} + \frac{704606643}{28107571549} a^{13} + \frac{475968940}{28107571549} a^{12} - \frac{389730326}{28107571549} a^{11} + \frac{48011578}{28107571549} a^{10} - \frac{6043594636}{28107571549} a^{9} - \frac{7191222613}{28107571549} a^{8} + \frac{4999953929}{28107571549} a^{7} + \frac{1536242570}{28107571549} a^{6} + \frac{7723780790}{28107571549} a^{5} - \frac{1611111986}{28107571549} a^{4} + \frac{13936125240}{28107571549} a^{3} + \frac{8103539217}{28107571549} a^{2} + \frac{8624444658}{28107571549} a + \frac{59099084}{1479345871}$, $\frac{1}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{24} - \frac{748994118072850482783341749939830868605606609622814165378191751284760429430422498687476407656951508239}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{23} - \frac{7532565446821145809820496997584472631154776064409145959480147054005142624403227987954519121172004839478624031}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{22} + \frac{1193398853033035413757676846708064451826157169384374902275280314154518311472135132729342772312894791341400556}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{21} + \frac{5470955398409043102262682287613933828556086381187334405867744103527015878048293611006981585598047298169904283}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{20} - \frac{216688736104461647172734824923022671983657595644478586536763896493389225486870520465783173307364263153419102033}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{19} - \frac{72580510824630349729042112098454071663385208072645488897777836200563419193887274352948394923298741045473155481}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{18} + \frac{184608839568689880769125738449981688960224596764842147961906144557697189200436341240326377283868655809033776957}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{17} - \frac{14540421576996095432051603505979855873721015447766061250561222147925961502870624134027309467266107704547906270}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{16} + \frac{3314635308649707558458215658844332813087562343501983807808077020149554118483427807790164218748203475256560900813}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{15} + \frac{2179879092809446049379607602122244728625558110386105482658103219214373768571508289895963774717258483169147198769}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{14} - \frac{153317319113915586246991417857105298750276971088710716345589025464559295037208425597035714252376683498692464115}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{13} - \frac{4458413114805438825147921310514845226129702500787950884303392417299929543749433426026747029091100288765816363403}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{12} + \frac{637513144652750147571808538255666128780655508596708280799949956008071243155635591778517910940308140654907629856}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{11} - \frac{4244373765604683312363655980610484048699715941388096159509529381070595767849237778737356044457544011318869210796}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{10} + \frac{6760024252460019353543985740544328063214994699343859103667699097235139545095378567439968038358298342510646001684}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{9} + \frac{59683750724943276931636064492248920057713702918712955139231472633691777830486093987101966990915581798527787452916}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{8} - \frac{69408351015817506467634660881856823216281248367284362598249413471696367130809087506253913881191099913834713552383}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{7} - \frac{34056868049814475912835599322732997937530655105669900017428287593622533266806528618163139861727279558553786486586}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{6} - \frac{68511462654482976450733336494882684281043223956147883174839159769775140207484331889649715515611528840620966333789}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{5} - \frac{10097994950836280058291215668118286670964139089253524103527713309276911154558664558138345042187260270124725825856}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{4} - \frac{50185109381512271520237968795153552744345131542961384969591165111012210803212433128652931918430276873611988894876}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a^{3} + \frac{3258104578383781218465242042645929123935921884535308322232098680960219289218231363570594228135600999585938215474}{9372298119843838271863911360405967442527795016019079740265517297032137911693913610395037992317749439940739040141} a^{2} + \frac{67947002812157077732570314231055851356236995289540832721734377382510240484859457537244476514054287895131644863108}{178073664277032927165414315847713381408028105304362515065044828643610620322184358597505721854037239358874041762679} a - \frac{1545251775571214443767873067698247952285666633495461976891377757641503775134832748600578265610826905684590911812}{9372298119843838271863911360405967442527795016019079740265517297032137911693913610395037992317749439940739040141}$

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:  $24$
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:  \( 511722181345293600000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 25
The 25 conjugacy class representatives for $C_{25}$
Character table for $C_{25}$ is not computed

Intermediate fields

5.5.241474942801.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ $25$ $25$

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
701Data not computed