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Properties

Label 25.25.6109899266...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 31^{20}$
Root discriminant $204.85$
Ramified primes $5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11035, -581975, 650425, 12227675, -7421440, -81703398, 51074860, 224926530, -172562760, -242987100, 219291821, 106088075, -120156280, -18329830, 33718945, -46803, -5240530, 442680, 461155, -61675, -22336, 3680, 540, -100, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 100*x^23 + 540*x^22 + 3680*x^21 - 22336*x^20 - 61675*x^19 + 461155*x^18 + 442680*x^17 - 5240530*x^16 - 46803*x^15 + 33718945*x^14 - 18329830*x^13 - 120156280*x^12 + 106088075*x^11 + 219291821*x^10 - 242987100*x^9 - 172562760*x^8 + 224926530*x^7 + 51074860*x^6 - 81703398*x^5 - 7421440*x^4 + 12227675*x^3 + 650425*x^2 - 581975*x - 11035)
 
gp: K = bnfinit(x^25 - 5*x^24 - 100*x^23 + 540*x^22 + 3680*x^21 - 22336*x^20 - 61675*x^19 + 461155*x^18 + 442680*x^17 - 5240530*x^16 - 46803*x^15 + 33718945*x^14 - 18329830*x^13 - 120156280*x^12 + 106088075*x^11 + 219291821*x^10 - 242987100*x^9 - 172562760*x^8 + 224926530*x^7 + 51074860*x^6 - 81703398*x^5 - 7421440*x^4 + 12227675*x^3 + 650425*x^2 - 581975*x - 11035, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 100 x^{23} + 540 x^{22} + 3680 x^{21} - 22336 x^{20} - 61675 x^{19} + 461155 x^{18} + 442680 x^{17} - 5240530 x^{16} - 46803 x^{15} + 33718945 x^{14} - 18329830 x^{13} - 120156280 x^{12} + 106088075 x^{11} + 219291821 x^{10} - 242987100 x^{9} - 172562760 x^{8} + 224926530 x^{7} + 51074860 x^{6} - 81703398 x^{5} - 7421440 x^{4} + 12227675 x^{3} + 650425 x^{2} - 581975 x - 11035 \)

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:  \(6109899266621889928830047225355883711017668247222900390625=5^{40}\cdot 31^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.85$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(775=5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{775}(256,·)$, $\chi_{775}(1,·)$, $\chi_{775}(66,·)$, $\chi_{775}(326,·)$, $\chi_{775}(591,·)$, $\chi_{775}(16,·)$, $\chi_{775}(721,·)$, $\chi_{775}(466,·)$, $\chi_{775}(531,·)$, $\chi_{775}(281,·)$, $\chi_{775}(411,·)$, $\chi_{775}(156,·)$, $\chi_{775}(221,·)$, $\chi_{775}(481,·)$, $\chi_{775}(101,·)$, $\chi_{775}(746,·)$, $\chi_{775}(171,·)$, $\chi_{775}(621,·)$, $\chi_{775}(686,·)$, $\chi_{775}(436,·)$, $\chi_{775}(566,·)$, $\chi_{775}(311,·)$, $\chi_{775}(376,·)$, $\chi_{775}(636,·)$, $\chi_{775}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{350} a^{20} - \frac{4}{35} a^{19} - \frac{1}{70} a^{18} - \frac{1}{7} a^{17} + \frac{1}{70} a^{16} - \frac{87}{350} a^{15} - \frac{3}{70} a^{14} - \frac{8}{35} a^{13} - \frac{3}{70} a^{12} - \frac{1}{35} a^{11} + \frac{109}{350} a^{10} - \frac{1}{70} a^{9} - \frac{19}{70} a^{8} - \frac{13}{70} a^{6} - \frac{113}{350} a^{5} - \frac{2}{5} a^{4} + \frac{3}{14} a^{3} - \frac{1}{14} a^{2} - \frac{1}{7} a + \frac{3}{70}$, $\frac{1}{1050} a^{21} - \frac{41}{210} a^{19} - \frac{1}{14} a^{18} - \frac{7}{30} a^{17} + \frac{113}{1050} a^{16} - \frac{17}{105} a^{15} + \frac{2}{105} a^{14} - \frac{13}{210} a^{13} - \frac{26}{105} a^{12} + \frac{59}{1050} a^{11} + \frac{31}{210} a^{10} - \frac{47}{105} a^{9} - \frac{5}{42} a^{8} - \frac{8}{35} a^{7} - \frac{44}{525} a^{6} + \frac{13}{210} a^{5} + \frac{1}{14} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{29}{105} a + \frac{17}{42}$, $\frac{1}{31500} a^{22} - \frac{2}{7875} a^{21} + \frac{11}{31500} a^{20} + \frac{71}{315} a^{19} + \frac{223}{1260} a^{18} + \frac{361}{1750} a^{17} - \frac{233}{1750} a^{16} - \frac{192}{875} a^{15} + \frac{13}{75} a^{14} - \frac{127}{3150} a^{13} + \frac{811}{3500} a^{12} - \frac{488}{7875} a^{11} - \frac{1599}{3500} a^{10} + \frac{52}{225} a^{9} + \frac{649}{3150} a^{8} - \frac{2167}{7875} a^{7} + \frac{2027}{15750} a^{6} + \frac{3368}{7875} a^{5} + \frac{1501}{3150} a^{4} - \frac{13}{70} a^{3} + \frac{187}{450} a^{2} + \frac{389}{1575} a + \frac{1753}{6300}$, $\frac{1}{189000} a^{23} - \frac{1}{189000} a^{22} + \frac{1}{4200} a^{21} + \frac{247}{189000} a^{20} - \frac{433}{2520} a^{19} + \frac{5989}{27000} a^{18} + \frac{1069}{10500} a^{17} + \frac{1}{84} a^{16} + \frac{1531}{31500} a^{15} - \frac{74}{945} a^{14} + \frac{6137}{27000} a^{13} + \frac{19441}{189000} a^{12} - \frac{6439}{37800} a^{11} - \frac{127}{189000} a^{10} + \frac{83}{315} a^{9} + \frac{3727}{31500} a^{8} + \frac{2647}{7875} a^{7} + \frac{307}{2100} a^{6} - \frac{207}{500} a^{5} + \frac{3667}{18900} a^{4} - \frac{649}{1350} a^{3} - \frac{2659}{18900} a^{2} + \frac{109}{280} a - \frac{7169}{37800}$, $\frac{1}{90881898471303024402608386807527688881437194816164112415053529466000} a^{24} - \frac{48841237542297851619004436655567626679206204290894263048509}{3029396615710100813420279560250922962714573160538803747168450982200} a^{23} - \frac{351920304957058593438516250832507253116074705610454565985394821}{45440949235651512201304193403763844440718597408082056207526764733000} a^{22} - \frac{1279644734679700338831873805338921420913040278915639706614188119}{9088189847130302440260838680752768888143719481616411241505352946600} a^{21} + \frac{2714081685901029371125634343110290661438307590834100449727346559}{6491564176521644600186313343394834920102656772583150886789537819000} a^{20} + \frac{5803325579639670863960187879580102992896109428381170310513287756049}{45440949235651512201304193403763844440718597408082056207526764733000} a^{19} + \frac{1496972081205585102254446227375778783564781255641814302805027949491}{18176379694260604880521677361505537776287438963232822483010705893200} a^{18} + \frac{22582885158341829561576694985666595710469903130991491895733208429}{420749529959736224086149938923739300377024050074833853773395969750} a^{17} - \frac{177171338966770727188866174752911108264546490293000688117000528917}{757349153927525203355069890062730740678643290134700936792112745550} a^{16} + \frac{6143814938895513514784221958013784457355156076779253546924410039259}{45440949235651512201304193403763844440718597408082056207526764733000} a^{15} + \frac{18916780996185412610399074458057658634518499982548829468376437361399}{90881898471303024402608386807527688881437194816164112415053529466000} a^{14} - \frac{174789041881544352930600784861981639512956497238136466452696371}{112199874655929659756306650379663813433873080019955694339572258600} a^{13} - \frac{528795899846877579128145898951850048491041095422991445399047487067}{22720474617825756100652096701881922220359298704041028103763382366500} a^{12} - \frac{2649640561071877123255162545797017605890672766877238787776164676}{14024984331991207469538331297457976679234135002494461792446532325} a^{11} - \frac{11644133316604514212135693962067514667587953736180225146566715141801}{90881898471303024402608386807527688881437194816164112415053529466000} a^{10} + \frac{368434389274288196891232818996442742251831131537168140632287525783}{1682998119838944896344599755694957201508096200299335415093583879000} a^{9} + \frac{1238794447614856546736748291557482547291289240788385176763995938867}{3029396615710100813420279560250922962714573160538803747168450982200} a^{8} + \frac{6832922970300792207305026401452079122409606270337983228242601336901}{15146983078550504067101397801254614813572865802694018735842254911000} a^{7} + \frac{4093546443784430592164655744804979394125420569057037504403872913}{20195977438067338756135197068339486418097154403592024981123006548} a^{6} + \frac{206000560611470956579514614332439364725764824524025261652428992379}{11360237308912878050326048350940961110179649352020514051881691183250} a^{5} - \frac{885434312681895046604220671786437488546053112575344132667344993931}{1817637969426060488052167736150553777628743896323282248301070589320} a^{4} - \frac{40658444963313444317955494960420173523568995814050277935395115953}{86554189020288594669150844578597798934702090301108678490527170920} a^{3} - \frac{3061868513831783509377744332007278070407208915969122784978967838951}{18176379694260604880521677361505537776287438963232822483010705893200} a^{2} + \frac{44726012258160048814565876482818509453238083029527027247258830071}{908818984713030244026083868075276888814371948161641124150535294660} a + \frac{2273018866284077278522211497604658640722163679367636558052442902573}{18176379694260604880521677361505537776287438963232822483010705893200}$

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:  \( 1887550003250961800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.390625.1, 5.5.360750390625.1, 5.5.360750390625.4, 5.5.360750390625.3, 5.5.360750390625.2, 5.5.923521.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
31Data not computed