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Properties

Label 25.25.126...401.1
Degree $25$
Signature $[25, 0]$
Discriminant $1.270\times 10^{48}$
Root discriminant $83.97$
Ramified prime $101$
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 - x^24 - 48*x^23 + 43*x^22 + 946*x^21 - 752*x^20 - 9993*x^19 + 6962*x^18 + 62052*x^17 - 37341*x^16 - 234195*x^15 + 119366*x^14 + 538390*x^13 - 226505*x^12 - 737819*x^11 + 249907*x^10 + 571793*x^9 - 151052*x^8 - 224456*x^7 + 42136*x^6 + 35494*x^5 - 2561*x^4 - 1633*x^3 + 57*x^2 + 19*x - 1)
 
gp: K = bnfinit(x^25 - x^24 - 48*x^23 + 43*x^22 + 946*x^21 - 752*x^20 - 9993*x^19 + 6962*x^18 + 62052*x^17 - 37341*x^16 - 234195*x^15 + 119366*x^14 + 538390*x^13 - 226505*x^12 - 737819*x^11 + 249907*x^10 + 571793*x^9 - 151052*x^8 - 224456*x^7 + 42136*x^6 + 35494*x^5 - 2561*x^4 - 1633*x^3 + 57*x^2 + 19*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 19, 57, -1633, -2561, 35494, 42136, -224456, -151052, 571793, 249907, -737819, -226505, 538390, 119366, -234195, -37341, 62052, 6962, -9993, -752, 946, 43, -48, -1, 1]);
 

\(x^{25} - x^{24} - 48 x^{23} + 43 x^{22} + 946 x^{21} - 752 x^{20} - 9993 x^{19} + 6962 x^{18} + 62052 x^{17} - 37341 x^{16} - 234195 x^{15} + 119366 x^{14} + 538390 x^{13} - 226505 x^{12} - 737819 x^{11} + 249907 x^{10} + 571793 x^{9} - 151052 x^{8} - 224456 x^{7} + 42136 x^{6} + 35494 x^{5} - 2561 x^{4} - 1633 x^{3} + 57 x^{2} + 19 x - 1\)  Toggle raw display

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:  \(1269734648531914468903714880493455422104626762401\)\(\medspace = 101^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $83.97$
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:  $101$
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:  \(101\)
Dirichlet character group:    $\lbrace$$\chi_{101}(1,·)$, $\chi_{101}(68,·)$, $\chi_{101}(5,·)$, $\chi_{101}(71,·)$, $\chi_{101}(78,·)$, $\chi_{101}(79,·)$, $\chi_{101}(16,·)$, $\chi_{101}(81,·)$, $\chi_{101}(19,·)$, $\chi_{101}(84,·)$, $\chi_{101}(87,·)$, $\chi_{101}(24,·)$, $\chi_{101}(25,·)$, $\chi_{101}(92,·)$, $\chi_{101}(31,·)$, $\chi_{101}(80,·)$, $\chi_{101}(36,·)$, $\chi_{101}(37,·)$, $\chi_{101}(97,·)$, $\chi_{101}(52,·)$, $\chi_{101}(54,·)$, $\chi_{101}(56,·)$, $\chi_{101}(88,·)$, $\chi_{101}(58,·)$, $\chi_{101}(95,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{809} a^{23} - \frac{315}{809} a^{22} + \frac{237}{809} a^{21} - \frac{290}{809} a^{20} + \frac{92}{809} a^{19} + \frac{157}{809} a^{18} + \frac{10}{809} a^{17} - \frac{88}{809} a^{16} - \frac{194}{809} a^{15} + \frac{162}{809} a^{14} + \frac{105}{809} a^{13} + \frac{333}{809} a^{12} - \frac{221}{809} a^{11} - \frac{126}{809} a^{10} - \frac{41}{809} a^{9} + \frac{366}{809} a^{8} + \frac{27}{809} a^{7} - \frac{136}{809} a^{6} - \frac{183}{809} a^{5} - \frac{129}{809} a^{4} + \frac{348}{809} a^{3} + \frac{100}{809} a^{2} - \frac{348}{809} a + \frac{133}{809}$, $\frac{1}{3924931990014460708094129751350317973059} a^{24} - \frac{312468728906810600397959776806806719}{3924931990014460708094129751350317973059} a^{23} + \frac{1539910052469863852062906239587702188335}{3924931990014460708094129751350317973059} a^{22} - \frac{1958814356801804597879029427540986313446}{3924931990014460708094129751350317973059} a^{21} + \frac{6377023749985007835794239821544685684}{13395672320868466580526040106997672263} a^{20} + \frac{324167581744813269420809625852756496767}{3924931990014460708094129751350317973059} a^{19} + \frac{459778914107668070929540000340987789687}{3924931990014460708094129751350317973059} a^{18} + \frac{929192756457990238355746804932501856458}{3924931990014460708094129751350317973059} a^{17} + \frac{192195572641135650378090775443484417260}{3924931990014460708094129751350317973059} a^{16} - \frac{1492141140751753791588026160190006035551}{3924931990014460708094129751350317973059} a^{15} + \frac{432474786473290779899323015529976275150}{3924931990014460708094129751350317973059} a^{14} + \frac{1909899722221996967526926974703720208687}{3924931990014460708094129751350317973059} a^{13} + \frac{1239237157944514323335423738684383391440}{3924931990014460708094129751350317973059} a^{12} + \frac{515716619115314781285187205986271725240}{3924931990014460708094129751350317973059} a^{11} - \frac{1325523699172288732854373410347205729350}{3924931990014460708094129751350317973059} a^{10} + \frac{681366914330040734466747029531644902698}{3924931990014460708094129751350317973059} a^{9} + \frac{380145157982285937088841752916872597394}{3924931990014460708094129751350317973059} a^{8} - \frac{1710236828687369467697485535676536493936}{3924931990014460708094129751350317973059} a^{7} - \frac{1583415115030320691262090557837009330215}{3924931990014460708094129751350317973059} a^{6} - \frac{1345069631769368109828031379540329089997}{3924931990014460708094129751350317973059} a^{5} - \frac{317066758514777751648368647709757773556}{3924931990014460708094129751350317973059} a^{4} + \frac{606081201141324543757269901260282925086}{3924931990014460708094129751350317973059} a^{3} + \frac{1561617130488087624579794765236937071340}{3924931990014460708094129751350317973059} a^{2} - \frac{1378953798207012513834470229548008415363}{3924931990014460708094129751350317973059} a + \frac{331233819462536232212091530392342656536}{3924931990014460708094129751350317973059}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 7091595602831931.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{25}\cdot(2\pi)^{0}\cdot 7091595602831931.0 \cdot 1}{2\sqrt{1269734648531914468903714880493455422104626762401}}\approx 0.105586251664491$ (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

5.5.104060401.1

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 $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$

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
101Data not computed