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Properties

Label 25.25.1638616440...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 41^{20}$
Root discriminant $256.20$
Ramified primes $5, 41$
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![3976207, -1592555, -124937795, -112036295, 1050477480, 1848883042, -1080207840, -3125972470, 209338160, 2251577330, 183446647, -883452625, -120400780, 207946490, 32118865, -30641007, -4699810, 2867330, 399035, -169085, -19326, 6060, 490, -120, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 120*x^23 + 490*x^22 + 6060*x^21 - 19326*x^20 - 169085*x^19 + 399035*x^18 + 2867330*x^17 - 4699810*x^16 - 30641007*x^15 + 32118865*x^14 + 207946490*x^13 - 120400780*x^12 - 883452625*x^11 + 183446647*x^10 + 2251577330*x^9 + 209338160*x^8 - 3125972470*x^7 - 1080207840*x^6 + 1848883042*x^5 + 1050477480*x^4 - 112036295*x^3 - 124937795*x^2 - 1592555*x + 3976207)
 
gp: K = bnfinit(x^25 - 5*x^24 - 120*x^23 + 490*x^22 + 6060*x^21 - 19326*x^20 - 169085*x^19 + 399035*x^18 + 2867330*x^17 - 4699810*x^16 - 30641007*x^15 + 32118865*x^14 + 207946490*x^13 - 120400780*x^12 - 883452625*x^11 + 183446647*x^10 + 2251577330*x^9 + 209338160*x^8 - 3125972470*x^7 - 1080207840*x^6 + 1848883042*x^5 + 1050477480*x^4 - 112036295*x^3 - 124937795*x^2 - 1592555*x + 3976207, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 120 x^{23} + 490 x^{22} + 6060 x^{21} - 19326 x^{20} - 169085 x^{19} + 399035 x^{18} + 2867330 x^{17} - 4699810 x^{16} - 30641007 x^{15} + 32118865 x^{14} + 207946490 x^{13} - 120400780 x^{12} - 883452625 x^{11} + 183446647 x^{10} + 2251577330 x^{9} + 209338160 x^{8} - 3125972470 x^{7} - 1080207840 x^{6} + 1848883042 x^{5} + 1050477480 x^{4} - 112036295 x^{3} - 124937795 x^{2} - 1592555 x + 3976207 \)

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:  \(1638616440290397163180658445206072428845800459384918212890625=5^{40}\cdot 41^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.20$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1025=5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1025}(256,·)$, $\chi_{1025}(1,·)$, $\chi_{1025}(836,·)$, $\chi_{1025}(961,·)$, $\chi_{1025}(201,·)$, $\chi_{1025}(141,·)$, $\chi_{1025}(206,·)$, $\chi_{1025}(461,·)$, $\chi_{1025}(16,·)$, $\chi_{1025}(406,·)$, $\chi_{1025}(666,·)$, $\chi_{1025}(411,·)$, $\chi_{1025}(346,·)$, $\chi_{1025}(611,·)$, $\chi_{1025}(551,·)$, $\chi_{1025}(616,·)$, $\chi_{1025}(426,·)$, $\chi_{1025}(871,·)$, $\chi_{1025}(221,·)$, $\chi_{1025}(816,·)$, $\chi_{1025}(51,·)$, $\chi_{1025}(756,·)$, $\chi_{1025}(821,·)$, $\chi_{1025}(631,·)$, $\chi_{1025}(1021,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{19} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{126} a^{20} - \frac{5}{126} a^{19} + \frac{1}{18} a^{18} - \frac{1}{63} a^{17} + \frac{4}{63} a^{16} - \frac{1}{14} a^{15} - \frac{8}{63} a^{14} - \frac{4}{21} a^{12} - \frac{19}{126} a^{11} + \frac{29}{63} a^{10} + \frac{31}{63} a^{9} + \frac{17}{63} a^{8} - \frac{47}{126} a^{7} + \frac{25}{63} a^{6} + \frac{1}{7} a^{5} - \frac{4}{9} a^{4} + \frac{3}{14} a^{3} - \frac{1}{18} a^{2} + \frac{11}{63} a - \frac{20}{63}$, $\frac{1}{126} a^{21} + \frac{1}{42} a^{19} - \frac{1}{14} a^{18} - \frac{1}{63} a^{17} + \frac{5}{63} a^{16} + \frac{1}{63} a^{15} + \frac{25}{126} a^{14} - \frac{4}{21} a^{13} + \frac{29}{126} a^{12} - \frac{8}{63} a^{11} + \frac{29}{63} a^{10} - \frac{13}{126} a^{9} - \frac{5}{14} a^{8} - \frac{19}{63} a^{7} - \frac{47}{126} a^{6} + \frac{55}{126} a^{5} + \frac{10}{63} a^{4} - \frac{19}{126} a^{3} + \frac{29}{126} a^{2} + \frac{2}{9} a - \frac{11}{126}$, $\frac{1}{252} a^{22} - \frac{1}{252} a^{20} - \frac{5}{126} a^{19} - \frac{1}{28} a^{18} + \frac{1}{14} a^{17} + \frac{1}{21} a^{16} - \frac{1}{126} a^{15} - \frac{11}{63} a^{14} - \frac{17}{126} a^{13} + \frac{59}{252} a^{12} + \frac{25}{126} a^{11} - \frac{11}{36} a^{10} + \frac{53}{126} a^{9} - \frac{5}{14} a^{8} - \frac{4}{21} a^{7} - \frac{41}{126} a^{6} - \frac{13}{63} a^{5} + \frac{25}{63} a^{4} + \frac{13}{126} a^{3} - \frac{4}{9} a^{2} + \frac{4}{21} a + \frac{97}{252}$, $\frac{1}{504} a^{23} - \frac{1}{504} a^{22} + \frac{1}{504} a^{21} + \frac{1}{504} a^{20} + \frac{41}{504} a^{19} - \frac{5}{504} a^{18} + \frac{1}{42} a^{17} + \frac{1}{252} a^{16} - \frac{1}{252} a^{15} + \frac{17}{126} a^{14} - \frac{9}{56} a^{13} - \frac{23}{504} a^{12} + \frac{113}{504} a^{11} + \frac{55}{168} a^{10} + \frac{13}{63} a^{9} - \frac{53}{126} a^{8} + \frac{67}{252} a^{7} + \frac{25}{126} a^{6} - \frac{5}{126} a^{5} + \frac{1}{14} a^{4} - \frac{79}{252} a^{3} + \frac{8}{63} a^{2} - \frac{53}{504} a - \frac{61}{168}$, $\frac{1}{434947484164575549910078340455335184114542961016767172360356413256078107464385450192} a^{24} + \frac{8311315996259933346033179236623923352070080625580358751963532411428386845295319}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{23} + \frac{19389096767740322948404775967542975245584401669411975172781848723270436531811387}{24163749120254197217226574469740843561919053389820398464464245180893228192465858344} a^{22} + \frac{19674541515316250377572506115478941448078422636132872871780308720568569532267931}{9061405920095323956459965426152816335719645021182649424174091942834960572174696879} a^{21} + \frac{104151932151980701469554388338362858508777513234918847413773602551700894772780967}{54368435520571943738759792556916898014317870127095896545044551657009763433048181274} a^{20} - \frac{17058953841975681554950323761368254232844412101438656617376290450146710907259070801}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{19} - \frac{31115065264386300264583424151986435202322161979287163671064349040300422431737361495}{434947484164575549910078340455335184114542961016767172360356413256078107464385450192} a^{18} + \frac{810943673645568516168148694878198548001971057674489814841176480952347498032066015}{72491247360762591651679723409222530685757160169461195393392735542679684577397575032} a^{17} - \frac{195727598418659090070413108641793483881703842756726174968511024042342293686976971}{3883459680040853124197128039779778429594133580506849753217467975500697388074870091} a^{16} - \frac{1653304022954583415089843315397835875173119037769018997180036260643801483873774597}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{15} - \frac{1220553541037418410644789883944092301816053883496772946803882567988973286769903603}{62135354880653649987154048636476454873506137288109596051479487608011158209197921456} a^{14} - \frac{41613475124519268549522014075712674297570779275585568575977392796247150032727591719}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{13} - \frac{622161120037002966526781227677339838046281386794043228331855997765353040973580925}{15533838720163412496788512159119113718376534322027399012869871902002789552299480364} a^{12} - \frac{1596522239449523374444511712816786230735954809031960800391489065108473458324965775}{36245623680381295825839861704611265342878580084730597696696367771339842288698787516} a^{11} + \frac{175841400013139890364830567071604509892994482881543228900741792527476477087701818183}{434947484164575549910078340455335184114542961016767172360356413256078107464385450192} a^{10} + \frac{7178382491483917590361431872087880328366687820937436698123419970120030960076944653}{27184217760285971869379896278458449007158935063547948272522275828504881716524090637} a^{9} + \frac{24794132929027945765824160494686477165819289197675456562318585476238969723060469445}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{8} + \frac{72905251820542555950177728946244545086963224548767448564935515222630140260697739723}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a^{7} - \frac{49326421566271633486162523722739637613527841282946234374102018075700339942651184925}{108736871041143887477519585113833796028635740254191793090089103314019526866096362548} a^{6} - \frac{7703143947311420574414589106359371291823665794530932840848412899514058796406652403}{36245623680381295825839861704611265342878580084730597696696367771339842288698787516} a^{5} - \frac{23188404859789262892005416435307144352827881215234102752256761166711625266439387219}{72491247360762591651679723409222530685757160169461195393392735542679684577397575032} a^{4} + \frac{36117082921357427690124735770624963973330426197681568037001583998887146862368305483}{72491247360762591651679723409222530685757160169461195393392735542679684577397575032} a^{3} - \frac{213962757666341018131576848419361705670475936463189635332966405466815188832952339249}{434947484164575549910078340455335184114542961016767172360356413256078107464385450192} a^{2} + \frac{57405756571925350682462397494348275417032490651324802202252412980567236574260429549}{217473742082287774955039170227667592057271480508383586180178206628039053732192725096} a - \frac{130998807557892696519860702382368008192568550099263694173341874574272322371581095001}{434947484164575549910078340455335184114542961016767172360356413256078107464385450192}$

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:  \( 8051564073328206000000 \) (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, 5.5.1103812890625.3, 5.5.1103812890625.4, 5.5.2825761.1, 5.5.1103812890625.2, 5.5.1103812890625.1

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
5Data not computed
41Data not computed