Properties

Label 25.25.611...625.1
Degree $25$
Signature $[25, 0]$
Discriminant $6.119\times 10^{48}$
Root discriminant $89.43$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5^2$ (as 25T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 60*x^23 + 290*x^22 + 1490*x^21 - 6836*x^20 - 20165*x^19 + 85905*x^18 + 165640*x^17 - 634300*x^16 - 876915*x^15 + 2860245*x^14 + 3104180*x^13 - 7920370*x^12 - 7429455*x^11 + 13143679*x^10 + 11662900*x^9 - 12187240*x^8 - 11034960*x^7 + 5336420*x^6 + 5352834*x^5 - 670950*x^4 - 973795*x^3 - 7365*x^2 + 53905*x + 4751)
 
gp: K = bnfinit(x^25 - 5*x^24 - 60*x^23 + 290*x^22 + 1490*x^21 - 6836*x^20 - 20165*x^19 + 85905*x^18 + 165640*x^17 - 634300*x^16 - 876915*x^15 + 2860245*x^14 + 3104180*x^13 - 7920370*x^12 - 7429455*x^11 + 13143679*x^10 + 11662900*x^9 - 12187240*x^8 - 11034960*x^7 + 5336420*x^6 + 5352834*x^5 - 670950*x^4 - 973795*x^3 - 7365*x^2 + 53905*x + 4751, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4751, 53905, -7365, -973795, -670950, 5352834, 5336420, -11034960, -12187240, 11662900, 13143679, -7429455, -7920370, 3104180, 2860245, -876915, -634300, 165640, 85905, -20165, -6836, 1490, 290, -60, -5, 1]);
 

\( x^{25} - 5 x^{24} - 60 x^{23} + 290 x^{22} + 1490 x^{21} - 6836 x^{20} - 20165 x^{19} + 85905 x^{18} + 165640 x^{17} - 634300 x^{16} - 876915 x^{15} + 2860245 x^{14} + 3104180 x^{13} - 7920370 x^{12} - 7429455 x^{11} + 13143679 x^{10} + 11662900 x^{9} - 12187240 x^{8} - 11034960 x^{7} + 5336420 x^{6} + 5352834 x^{5} - 670950 x^{4} - 973795 x^{3} - 7365 x^{2} + 53905 x + 4751 \)

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:  \(6118625560089276488733958103694021701812744140625\)\(\medspace = 5^{40}\cdot 11^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $89.43$
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:  $5, 11$
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:  \(275=5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{275}(256,·)$, $\chi_{275}(1,·)$, $\chi_{275}(196,·)$, $\chi_{275}(71,·)$, $\chi_{275}(136,·)$, $\chi_{275}(201,·)$, $\chi_{275}(141,·)$, $\chi_{275}(16,·)$, $\chi_{275}(81,·)$, $\chi_{275}(146,·)$, $\chi_{275}(86,·)$, $\chi_{275}(26,·)$, $\chi_{275}(91,·)$, $\chi_{275}(221,·)$, $\chi_{275}(31,·)$, $\chi_{275}(36,·)$, $\chi_{275}(166,·)$, $\chi_{275}(236,·)$, $\chi_{275}(111,·)$, $\chi_{275}(181,·)$, $\chi_{275}(246,·)$, $\chi_{275}(56,·)$, $\chi_{275}(251,·)$, $\chi_{275}(126,·)$, $\chi_{275}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{602} a^{20} + \frac{51}{602} a^{19} + \frac{13}{602} a^{18} - \frac{10}{301} a^{17} + \frac{53}{301} a^{16} + \frac{15}{301} a^{15} + \frac{18}{301} a^{14} + \frac{11}{602} a^{13} + \frac{1}{301} a^{12} - \frac{68}{301} a^{11} - \frac{229}{602} a^{10} + \frac{88}{301} a^{9} - \frac{29}{301} a^{8} - \frac{34}{301} a^{7} + \frac{26}{301} a^{6} - \frac{45}{301} a^{5} + \frac{223}{602} a^{4} - \frac{87}{301} a^{3} + \frac{123}{301} a^{2} + \frac{46}{301} a - \frac{45}{301}$, $\frac{1}{602} a^{21} + \frac{121}{602} a^{19} - \frac{81}{602} a^{18} - \frac{39}{301} a^{17} + \frac{3}{43} a^{16} + \frac{11}{602} a^{15} - \frac{19}{602} a^{14} + \frac{1}{14} a^{13} + \frac{9}{86} a^{12} + \frac{85}{602} a^{11} - \frac{185}{602} a^{10} + \frac{297}{602} a^{9} + \frac{181}{602} a^{8} + \frac{209}{602} a^{7} - \frac{33}{602} a^{6} - \frac{3}{602} a^{5} - \frac{109}{602} a^{4} + \frac{45}{301} a^{3} + \frac{94}{301} a^{2} + \frac{17}{301} a + \frac{75}{602}$, $\frac{1}{1204} a^{22} - \frac{1}{1204} a^{20} + \frac{9}{602} a^{19} - \frac{159}{1204} a^{18} + \frac{37}{602} a^{17} + \frac{11}{602} a^{16} + \frac{117}{602} a^{15} + \frac{83}{602} a^{14} + \frac{113}{602} a^{13} - \frac{159}{1204} a^{12} - \frac{37}{301} a^{11} - \frac{59}{1204} a^{10} - \frac{261}{602} a^{9} - \frac{60}{301} a^{8} + \frac{34}{301} a^{7} + \frac{144}{301} a^{6} - \frac{19}{86} a^{5} + \frac{144}{301} a^{4} + \frac{173}{602} a^{3} + \frac{61}{602} a^{2} - \frac{3}{301} a + \frac{445}{1204}$, $\frac{1}{2408} a^{23} - \frac{1}{2408} a^{22} + \frac{1}{2408} a^{21} + \frac{1}{2408} a^{20} + \frac{351}{2408} a^{19} - \frac{163}{2408} a^{18} - \frac{225}{1204} a^{17} + \frac{97}{1204} a^{16} - \frac{293}{1204} a^{15} + \frac{289}{1204} a^{14} - \frac{71}{344} a^{13} - \frac{501}{2408} a^{12} + \frac{299}{2408} a^{11} + \frac{881}{2408} a^{10} + \frac{29}{602} a^{9} - \frac{3}{301} a^{8} - \frac{116}{301} a^{7} + \frac{141}{602} a^{6} - \frac{289}{602} a^{5} - \frac{31}{301} a^{4} + \frac{39}{1204} a^{3} - \frac{41}{172} a^{2} - \frac{529}{2408} a + \frac{121}{2408}$, $\frac{1}{324009618705367518748020508096277676886098757887125488} a^{24} + \frac{13112345802792722735171697963681671809039187464611}{81002404676341879687005127024069419221524689471781372} a^{23} - \frac{956790993846792320067830999103287358658762280369}{20250601169085469921751281756017354805381172367945343} a^{22} - \frac{24143535404260328236853873926295217336434253734591}{162004809352683759374010254048138838443049378943562744} a^{21} + \frac{16638400572223579001652804812854918359918577223523}{81002404676341879687005127024069419221524689471781372} a^{20} - \frac{2408361540165019115189941011789414650306560169555741}{40501202338170939843502563512034709610762344735890686} a^{19} - \frac{47650867487131806842281424475058052156438933743161}{1076443915964676142020001688027500587661457667399088} a^{18} - \frac{15948871944938345513271737908371033513083293827074143}{81002404676341879687005127024069419221524689471781372} a^{17} + \frac{96551152201112813780955559098663063345981419648711}{1883776852938183248535002954048126028407550917948404} a^{16} + \frac{816550910616593948542028019684163248146575164976594}{20250601169085469921751281756017354805381172367945343} a^{15} - \frac{1620484022219547714716378739489900421840440909755969}{7535107411752732994140011816192504113630203671793616} a^{14} + \frac{25490226120719695527281341987100814673928179601457241}{162004809352683759374010254048138838443049378943562744} a^{13} - \frac{36966953343822458226236072212864316096643033282314525}{162004809352683759374010254048138838443049378943562744} a^{12} - \frac{19315778268948211996272601634230118896873296091750757}{81002404676341879687005127024069419221524689471781372} a^{11} + \frac{29517807113712577901621882238591900507258065171926677}{324009618705367518748020508096277676886098757887125488} a^{10} - \frac{6615332300506636928102388744109754038410410284826915}{81002404676341879687005127024069419221524689471781372} a^{9} + \frac{23942237796856723621735962487749510337086240828195}{941888426469091624267501477024063014203775458974202} a^{8} + \frac{18496321254252796765046978061766310267167523253413409}{40501202338170939843502563512034709610762344735890686} a^{7} - \frac{1394584138037812150229235091632911559659671564678762}{20250601169085469921751281756017354805381172367945343} a^{6} + \frac{34892194039374000680568574591348473702519774705868305}{81002404676341879687005127024069419221524689471781372} a^{5} + \frac{1683591901536520776800051688351798291431994641867827}{162004809352683759374010254048138838443049378943562744} a^{4} + \frac{843310366743188482845490089208952882580003866451634}{2892943024155067131678754536573907829340167481135049} a^{3} + \frac{47543337685782943597240781671339006029672303135758261}{324009618705367518748020508096277676886098757887125488} a^{2} + \frac{19169133585049232963097540114278090853846928790279173}{40501202338170939843502563512034709610762344735890686} a - \frac{4514502090012518523930413560450270313958726818658537}{46287088386481074106860072585182525269442679698160784}$

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:  \( 33431232217842756 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{25}\cdot(2\pi)^{0}\cdot 33431232217842756 \cdot 1}{2\sqrt{6118625560089276488733958103694021701812744140625}}\approx 0.226748979868915$ (assuming GRH)

Galois group

$C_5^2$ (as 25T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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, 5.5.5719140625.1, 5.5.5719140625.3, 5.5.5719140625.2, \(\Q(\zeta_{11})^+\), 5.5.5719140625.4

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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

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
11Data not computed