Properties

Label 25.25.338...625.1
Degree $25$
Signature $[25, 0]$
Discriminant $3.388\times 10^{47}$
Root discriminant $79.65$
Ramified prime $5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{25}$ (as 25T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 50*x^23 + 1025*x^21 - 11250*x^19 - 125*x^18 + 72525*x^17 + 3100*x^16 - 283885*x^15 - 28375*x^14 + 674550*x^13 + 121800*x^12 - 942450*x^11 - 261005*x^10 + 718625*x^9 + 269475*x^8 - 258425*x^7 - 117125*x^6 + 33010*x^5 + 16625*x^4 - 1100*x^3 - 650*x^2 - 50*x - 1)
 
gp: K = bnfinit(x^25 - 50*x^23 + 1025*x^21 - 11250*x^19 - 125*x^18 + 72525*x^17 + 3100*x^16 - 283885*x^15 - 28375*x^14 + 674550*x^13 + 121800*x^12 - 942450*x^11 - 261005*x^10 + 718625*x^9 + 269475*x^8 - 258425*x^7 - 117125*x^6 + 33010*x^5 + 16625*x^4 - 1100*x^3 - 650*x^2 - 50*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -50, -650, -1100, 16625, 33010, -117125, -258425, 269475, 718625, -261005, -942450, 121800, 674550, -28375, -283885, 3100, 72525, -125, -11250, 0, 1025, 0, -50, 0, 1]);
 

\( x^{25} - 50 x^{23} + 1025 x^{21} - 11250 x^{19} - 125 x^{18} + 72525 x^{17} + 3100 x^{16} - 283885 x^{15} - 28375 x^{14} + 674550 x^{13} + 121800 x^{12} - 942450 x^{11} - 261005 x^{10} + 718625 x^{9} + 269475 x^{8} - 258425 x^{7} - 117125 x^{6} + 33010 x^{5} + 16625 x^{4} - 1100 x^{3} - 650 x^{2} - 50 x - 1 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[25, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(338813178901720135627329000271856784820556640625\)\(\medspace = 5^{68}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $79.65$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $25$
This field is Galois and abelian over $\Q$.
Conductor:  \(125=5^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{125}(1,·)$, $\chi_{125}(66,·)$, $\chi_{125}(6,·)$, $\chi_{125}(71,·)$, $\chi_{125}(11,·)$, $\chi_{125}(76,·)$, $\chi_{125}(16,·)$, $\chi_{125}(81,·)$, $\chi_{125}(21,·)$, $\chi_{125}(86,·)$, $\chi_{125}(26,·)$, $\chi_{125}(91,·)$, $\chi_{125}(31,·)$, $\chi_{125}(96,·)$, $\chi_{125}(36,·)$, $\chi_{125}(101,·)$, $\chi_{125}(41,·)$, $\chi_{125}(106,·)$, $\chi_{125}(46,·)$, $\chi_{125}(111,·)$, $\chi_{125}(51,·)$, $\chi_{125}(116,·)$, $\chi_{125}(56,·)$, $\chi_{125}(121,·)$, $\chi_{125}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{48443} a^{23} - \frac{8914}{48443} a^{22} - \frac{620}{48443} a^{21} + \frac{13840}{48443} a^{20} + \frac{19910}{48443} a^{19} - \frac{5665}{48443} a^{18} - \frac{4888}{48443} a^{17} - \frac{7999}{48443} a^{16} + \frac{5969}{48443} a^{15} - \frac{3058}{48443} a^{14} + \frac{3253}{48443} a^{13} - \frac{18499}{48443} a^{12} + \frac{653}{48443} a^{11} - \frac{19}{48443} a^{10} - \frac{10051}{48443} a^{9} + \frac{17976}{48443} a^{8} - \frac{6864}{48443} a^{7} + \frac{6278}{48443} a^{6} - \frac{17208}{48443} a^{5} + \frac{23496}{48443} a^{4} + \frac{23755}{48443} a^{3} - \frac{21505}{48443} a^{2} - \frac{19512}{48443} a + \frac{18911}{48443}$, $\frac{1}{939614457656128672722660751801576776443} a^{24} + \frac{2208671930602834541315852043369777}{939614457656128672722660751801576776443} a^{23} - \frac{445856917464052609043804055864261185541}{939614457656128672722660751801576776443} a^{22} + \frac{316847791412327049866719576806175102083}{939614457656128672722660751801576776443} a^{21} - \frac{228354781568471371212343056096758725436}{939614457656128672722660751801576776443} a^{20} + \frac{274138170154871506019712788286411158254}{939614457656128672722660751801576776443} a^{19} + \frac{186873890186705868615473673448229529452}{939614457656128672722660751801576776443} a^{18} - \frac{371107551704899082527747529097449125840}{939614457656128672722660751801576776443} a^{17} + \frac{50662207847557522105671813632103828642}{939614457656128672722660751801576776443} a^{16} - \frac{83970657139526369542637251456689405582}{939614457656128672722660751801576776443} a^{15} + \frac{91592491938221242860581934288678337250}{939614457656128672722660751801576776443} a^{14} - \frac{28082399223187915929349037674785984072}{939614457656128672722660751801576776443} a^{13} - \frac{89048376692509386181452678544023093911}{939614457656128672722660751801576776443} a^{12} + \frac{419970588698717848022015765066201107358}{939614457656128672722660751801576776443} a^{11} + \frac{64515609990965187177519011765340294950}{939614457656128672722660751801576776443} a^{10} + \frac{33485471872995534021957304472335699695}{939614457656128672722660751801576776443} a^{9} + \frac{444965568721767485563473600905679769612}{939614457656128672722660751801576776443} a^{8} + \frac{313541698658954916103383017345359934962}{939614457656128672722660751801576776443} a^{7} + \frac{59380587804770555742648656190594523334}{939614457656128672722660751801576776443} a^{6} - \frac{394233703010706138483127208037309275}{3060633412560679715708992676878100249} a^{5} - \frac{270498902170414391382684053738187573758}{939614457656128672722660751801576776443} a^{4} + \frac{314864478514818487709163087750354843823}{939614457656128672722660751801576776443} a^{3} + \frac{216738722359055332867642428542718772419}{939614457656128672722660751801576776443} a^{2} + \frac{357692355498757990998870524111377297026}{939614457656128672722660751801576776443} a + \frac{202743076695873548614388376309087464644}{939614457656128672722660751801576776443}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $24$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5798047530670207.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{25}\cdot(2\pi)^{0}\cdot 5798047530670207.0 \cdot 1}{2\sqrt{338813178901720135627329000271856784820556640625}}\approx 0.167117342071058$ (assuming GRH)

Galois group

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

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

Intermediate fields

5.5.390625.1

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 $25$ $25$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ $25$ $25$ $25$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed