↑

Properties

Label 25.1.145...641.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.459\times 10^{32}$
Root discriminant \(19.34\)
Ramified prime $479$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{25}$ (as 25T4)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1)
 
gp: K = bnfinit(y^25 - 3*y^24 + 9*y^23 - 22*y^22 + 41*y^21 - 60*y^20 + 66*y^19 - 47*y^18 + 6*y^17 + 48*y^16 - 82*y^15 + 76*y^14 + 11*y^13 - 138*y^12 + 280*y^11 - 336*y^10 + 317*y^9 - 205*y^8 + 144*y^7 - 109*y^6 + 126*y^5 - 104*y^4 + 76*y^3 - 23*y^2 + 10*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1)
 

\( x^{25} - 3 x^{24} + 9 x^{23} - 22 x^{22} + 41 x^{21} - 60 x^{20} + 66 x^{19} - 47 x^{18} + 6 x^{17} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(145890213878661931676924574560641\) \(\medspace = 479^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(19.34\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $479^{1/2}\approx 21.88606862823929$
Ramified primes:   \(479\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{403}a^{22}+\frac{183}{403}a^{21}+\frac{8}{31}a^{20}-\frac{119}{403}a^{19}-\frac{129}{403}a^{18}-\frac{121}{403}a^{17}-\frac{120}{403}a^{16}+\frac{8}{31}a^{15}+\frac{82}{403}a^{14}-\frac{139}{403}a^{13}-\frac{144}{403}a^{12}-\frac{69}{403}a^{11}+\frac{127}{403}a^{10}+\frac{25}{403}a^{9}-\frac{38}{403}a^{8}+\frac{154}{403}a^{7}+\frac{198}{403}a^{6}-\frac{4}{403}a^{5}+\frac{166}{403}a^{4}+\frac{6}{13}a^{3}+\frac{140}{403}a^{2}-\frac{20}{403}a-\frac{171}{403}$, $\frac{1}{280891}a^{23}-\frac{19}{16523}a^{22}+\frac{111021}{280891}a^{21}-\frac{54355}{280891}a^{20}-\frac{60815}{280891}a^{19}+\frac{90945}{280891}a^{18}-\frac{101706}{280891}a^{17}+\frac{139409}{280891}a^{16}+\frac{86896}{280891}a^{15}-\frac{99663}{280891}a^{14}-\frac{133728}{280891}a^{13}+\frac{72392}{280891}a^{12}-\frac{3320}{16523}a^{11}+\frac{68350}{280891}a^{10}-\frac{4727}{21607}a^{9}-\frac{83383}{280891}a^{8}+\frac{17382}{280891}a^{7}+\frac{4360}{9061}a^{6}-\frac{131203}{280891}a^{5}+\frac{52807}{280891}a^{4}+\frac{80523}{280891}a^{3}-\frac{47083}{280891}a^{2}-\frac{1169}{16523}a-\frac{50091}{280891}$, $\frac{1}{45\!\cdots\!33}a^{24}-\frac{6541108081}{45\!\cdots\!33}a^{23}+\frac{1205898133468}{45\!\cdots\!33}a^{22}+\frac{17\!\cdots\!55}{45\!\cdots\!33}a^{21}-\frac{2507231126183}{111515200396613}a^{20}-\frac{361345718106582}{45\!\cdots\!33}a^{19}-\frac{864047747781664}{45\!\cdots\!33}a^{18}+\frac{15\!\cdots\!67}{45\!\cdots\!33}a^{17}+\frac{628616268405611}{45\!\cdots\!33}a^{16}+\frac{168813110239627}{45\!\cdots\!33}a^{15}-\frac{14\!\cdots\!86}{45\!\cdots\!33}a^{14}+\frac{15\!\cdots\!10}{45\!\cdots\!33}a^{13}+\frac{851364700477942}{45\!\cdots\!33}a^{12}+\frac{695491991136685}{45\!\cdots\!33}a^{11}-\frac{22\!\cdots\!11}{45\!\cdots\!33}a^{10}+\frac{13\!\cdots\!60}{45\!\cdots\!33}a^{9}-\frac{12\!\cdots\!38}{45\!\cdots\!33}a^{8}+\frac{855305453036068}{45\!\cdots\!33}a^{7}-\frac{20\!\cdots\!35}{45\!\cdots\!33}a^{6}-\frac{17\!\cdots\!42}{45\!\cdots\!33}a^{5}+\frac{19\!\cdots\!04}{45\!\cdots\!33}a^{4}-\frac{21\!\cdots\!73}{45\!\cdots\!33}a^{3}-\frac{142489639201522}{351701785866241}a^{2}-\frac{80694613063309}{45\!\cdots\!33}a-\frac{21\!\cdots\!81}{45\!\cdots\!33}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{291269505265016}{45\!\cdots\!33}a^{24}+\frac{46649143235665}{45\!\cdots\!33}a^{23}+\frac{630241046460984}{45\!\cdots\!33}a^{22}-\frac{400742235057064}{45\!\cdots\!33}a^{21}-\frac{20\!\cdots\!45}{45\!\cdots\!33}a^{20}+\frac{45\!\cdots\!91}{45\!\cdots\!33}a^{19}-\frac{87\!\cdots\!27}{45\!\cdots\!33}a^{18}+\frac{94\!\cdots\!77}{45\!\cdots\!33}a^{17}-\frac{41\!\cdots\!36}{45\!\cdots\!33}a^{16}-\frac{166769241844039}{45\!\cdots\!33}a^{15}+\frac{12\!\cdots\!12}{45\!\cdots\!33}a^{14}-\frac{10\!\cdots\!42}{351701785866241}a^{13}+\frac{19\!\cdots\!63}{45\!\cdots\!33}a^{12}+\frac{58\!\cdots\!63}{45\!\cdots\!33}a^{11}-\frac{18\!\cdots\!03}{45\!\cdots\!33}a^{10}+\frac{44\!\cdots\!52}{45\!\cdots\!33}a^{9}-\frac{28\!\cdots\!45}{351701785866241}a^{8}+\frac{28\!\cdots\!79}{45\!\cdots\!33}a^{7}+\frac{368553607867017}{351701785866241}a^{6}+\frac{93\!\cdots\!16}{45\!\cdots\!33}a^{5}-\frac{11\!\cdots\!26}{45\!\cdots\!33}a^{4}+\frac{425378108559064}{147487845685843}a^{3}-\frac{99\!\cdots\!38}{45\!\cdots\!33}a^{2}+\frac{984955107541420}{45\!\cdots\!33}a+\frac{37\!\cdots\!62}{45\!\cdots\!33}$, $\frac{394076129615867}{45\!\cdots\!33}a^{24}-\frac{243452668556596}{45\!\cdots\!33}a^{23}+\frac{13\!\cdots\!27}{45\!\cdots\!33}a^{22}-\frac{20\!\cdots\!88}{45\!\cdots\!33}a^{21}+\frac{498948696272092}{45\!\cdots\!33}a^{20}+\frac{21\!\cdots\!72}{45\!\cdots\!33}a^{19}-\frac{92\!\cdots\!60}{45\!\cdots\!33}a^{18}+\frac{13\!\cdots\!56}{45\!\cdots\!33}a^{17}-\frac{298734579588001}{111515200396613}a^{16}+\frac{86\!\cdots\!34}{45\!\cdots\!33}a^{15}+\frac{87\!\cdots\!69}{45\!\cdots\!33}a^{14}-\frac{16\!\cdots\!65}{45\!\cdots\!33}a^{13}+\frac{34\!\cdots\!62}{45\!\cdots\!33}a^{12}-\frac{12\!\cdots\!57}{351701785866241}a^{11}-\frac{140689766462529}{45\!\cdots\!33}a^{10}+\frac{42\!\cdots\!20}{45\!\cdots\!33}a^{9}-\frac{13\!\cdots\!94}{111515200396613}a^{8}+\frac{52\!\cdots\!85}{45\!\cdots\!33}a^{7}-\frac{14\!\cdots\!70}{45\!\cdots\!33}a^{6}+\frac{81\!\cdots\!57}{45\!\cdots\!33}a^{5}-\frac{61\!\cdots\!02}{45\!\cdots\!33}a^{4}+\frac{18\!\cdots\!82}{45\!\cdots\!33}a^{3}-\frac{25\!\cdots\!33}{45\!\cdots\!33}a^{2}+\frac{79\!\cdots\!27}{45\!\cdots\!33}a-\frac{31\!\cdots\!62}{45\!\cdots\!33}$, $\frac{885392108758130}{45\!\cdots\!33}a^{24}-\frac{19\!\cdots\!33}{45\!\cdots\!33}a^{23}+\frac{60\!\cdots\!63}{45\!\cdots\!33}a^{22}-\frac{13\!\cdots\!46}{45\!\cdots\!33}a^{21}+\frac{22\!\cdots\!16}{45\!\cdots\!33}a^{20}-\frac{28\!\cdots\!33}{45\!\cdots\!33}a^{19}+\frac{17\!\cdots\!29}{351701785866241}a^{18}-\frac{48\!\cdots\!01}{45\!\cdots\!33}a^{17}-\frac{18\!\cdots\!06}{45\!\cdots\!33}a^{16}+\frac{41\!\cdots\!42}{45\!\cdots\!33}a^{15}-\frac{30\!\cdots\!61}{351701785866241}a^{14}+\frac{18\!\cdots\!99}{45\!\cdots\!33}a^{13}+\frac{51\!\cdots\!29}{45\!\cdots\!33}a^{12}-\frac{10\!\cdots\!95}{45\!\cdots\!33}a^{11}+\frac{15\!\cdots\!06}{45\!\cdots\!33}a^{10}-\frac{12\!\cdots\!11}{45\!\cdots\!33}a^{9}+\frac{85\!\cdots\!04}{45\!\cdots\!33}a^{8}-\frac{12\!\cdots\!20}{45\!\cdots\!33}a^{7}+\frac{23\!\cdots\!67}{45\!\cdots\!33}a^{6}-\frac{29\!\cdots\!92}{45\!\cdots\!33}a^{5}+\frac{53\!\cdots\!27}{45\!\cdots\!33}a^{4}-\frac{30\!\cdots\!06}{45\!\cdots\!33}a^{3}+\frac{139703455524280}{111515200396613}a^{2}+\frac{12\!\cdots\!54}{45\!\cdots\!33}a-\frac{42\!\cdots\!16}{45\!\cdots\!33}$, $\frac{28869201169486}{147487845685843}a^{24}-\frac{11\!\cdots\!16}{45\!\cdots\!33}a^{23}+\frac{380106123328598}{351701785866241}a^{22}-\frac{93\!\cdots\!57}{45\!\cdots\!33}a^{21}+\frac{13\!\cdots\!65}{45\!\cdots\!33}a^{20}-\frac{14\!\cdots\!54}{45\!\cdots\!33}a^{19}+\frac{64\!\cdots\!03}{45\!\cdots\!33}a^{18}+\frac{62\!\cdots\!27}{45\!\cdots\!33}a^{17}-\frac{14\!\cdots\!70}{351701785866241}a^{16}+\frac{30\!\cdots\!66}{45\!\cdots\!33}a^{15}-\frac{14\!\cdots\!21}{45\!\cdots\!33}a^{14}+\frac{23\!\cdots\!04}{45\!\cdots\!33}a^{13}+\frac{57\!\cdots\!59}{45\!\cdots\!33}a^{12}-\frac{20\!\cdots\!81}{147487845685843}a^{11}+\frac{98\!\cdots\!51}{45\!\cdots\!33}a^{10}-\frac{37\!\cdots\!04}{45\!\cdots\!33}a^{9}+\frac{27\!\cdots\!43}{45\!\cdots\!33}a^{8}+\frac{36\!\cdots\!06}{45\!\cdots\!33}a^{7}+\frac{17\!\cdots\!95}{45\!\cdots\!33}a^{6}+\frac{89\!\cdots\!47}{45\!\cdots\!33}a^{5}+\frac{31\!\cdots\!15}{45\!\cdots\!33}a^{4}+\frac{96\!\cdots\!04}{45\!\cdots\!33}a^{3}-\frac{79\!\cdots\!58}{45\!\cdots\!33}a^{2}+\frac{13\!\cdots\!56}{45\!\cdots\!33}a-\frac{26\!\cdots\!01}{45\!\cdots\!33}$, $\frac{662800553483}{8675755628579}a^{24}-\frac{1722504568957}{268948424485949}a^{23}+\frac{24495478737982}{268948424485949}a^{22}+\frac{25953742598166}{268948424485949}a^{21}-\frac{9060952536553}{8675755628579}a^{20}+\frac{758055425489472}{268948424485949}a^{19}-\frac{13\!\cdots\!08}{268948424485949}a^{18}+\frac{16\!\cdots\!35}{268948424485949}a^{17}-\frac{11\!\cdots\!28}{268948424485949}a^{16}+\frac{121048187109474}{268948424485949}a^{15}+\frac{14\!\cdots\!42}{268948424485949}a^{14}-\frac{22\!\cdots\!57}{268948424485949}a^{13}+\frac{28\!\cdots\!55}{268948424485949}a^{12}-\frac{182813821234968}{268948424485949}a^{11}-\frac{25\!\cdots\!20}{268948424485949}a^{10}+\frac{70\!\cdots\!17}{268948424485949}a^{9}-\frac{547504718100486}{20688340345073}a^{8}+\frac{68\!\cdots\!81}{268948424485949}a^{7}-\frac{15\!\cdots\!50}{268948424485949}a^{6}+\frac{14\!\cdots\!95}{268948424485949}a^{5}-\frac{700864747780916}{268948424485949}a^{4}+\frac{32\!\cdots\!26}{268948424485949}a^{3}-\frac{15\!\cdots\!69}{268948424485949}a^{2}+\frac{14\!\cdots\!07}{268948424485949}a+\frac{625880206283273}{268948424485949}$, $\frac{838167706315892}{45\!\cdots\!33}a^{24}-\frac{20\!\cdots\!95}{45\!\cdots\!33}a^{23}+\frac{62\!\cdots\!50}{45\!\cdots\!33}a^{22}-\frac{35831493123845}{11345218898911}a^{21}+\frac{25\!\cdots\!44}{45\!\cdots\!33}a^{20}-\frac{34\!\cdots\!64}{45\!\cdots\!33}a^{19}+\frac{33\!\cdots\!30}{45\!\cdots\!33}a^{18}-\frac{643293624550450}{147487845685843}a^{17}-\frac{32\!\cdots\!61}{45\!\cdots\!33}a^{16}+\frac{31\!\cdots\!21}{45\!\cdots\!33}a^{15}-\frac{42\!\cdots\!25}{45\!\cdots\!33}a^{14}+\frac{26\!\cdots\!63}{351701785866241}a^{13}+\frac{27\!\cdots\!08}{45\!\cdots\!33}a^{12}-\frac{90\!\cdots\!92}{45\!\cdots\!33}a^{11}+\frac{16\!\cdots\!32}{45\!\cdots\!33}a^{10}-\frac{13\!\cdots\!49}{351701785866241}a^{9}+\frac{16\!\cdots\!37}{45\!\cdots\!33}a^{8}-\frac{94\!\cdots\!74}{45\!\cdots\!33}a^{7}+\frac{67\!\cdots\!50}{351701785866241}a^{6}-\frac{21\!\cdots\!42}{147487845685843}a^{5}+\frac{72\!\cdots\!04}{45\!\cdots\!33}a^{4}-\frac{41\!\cdots\!25}{45\!\cdots\!33}a^{3}+\frac{35\!\cdots\!32}{45\!\cdots\!33}a^{2}-\frac{15\!\cdots\!52}{45\!\cdots\!33}a+\frac{33\!\cdots\!54}{45\!\cdots\!33}$, $\frac{656837936489261}{45\!\cdots\!33}a^{24}-\frac{26\!\cdots\!82}{45\!\cdots\!33}a^{23}+\frac{79\!\cdots\!37}{45\!\cdots\!33}a^{22}-\frac{20\!\cdots\!70}{45\!\cdots\!33}a^{21}+\frac{41\!\cdots\!30}{45\!\cdots\!33}a^{20}-\frac{64\!\cdots\!97}{45\!\cdots\!33}a^{19}+\frac{77\!\cdots\!16}{45\!\cdots\!33}a^{18}-\frac{62\!\cdots\!10}{45\!\cdots\!33}a^{17}+\frac{17\!\cdots\!99}{45\!\cdots\!33}a^{16}+\frac{44\!\cdots\!00}{45\!\cdots\!33}a^{15}-\frac{94\!\cdots\!07}{45\!\cdots\!33}a^{14}+\frac{99\!\cdots\!70}{45\!\cdots\!33}a^{13}-\frac{23\!\cdots\!91}{45\!\cdots\!33}a^{12}-\frac{12\!\cdots\!60}{45\!\cdots\!33}a^{11}+\frac{29\!\cdots\!63}{45\!\cdots\!33}a^{10}-\frac{39\!\cdots\!89}{45\!\cdots\!33}a^{9}+\frac{38\!\cdots\!32}{45\!\cdots\!33}a^{8}-\frac{25\!\cdots\!45}{45\!\cdots\!33}a^{7}+\frac{14\!\cdots\!82}{45\!\cdots\!33}a^{6}-\frac{26\!\cdots\!43}{111515200396613}a^{5}+\frac{13\!\cdots\!71}{45\!\cdots\!33}a^{4}-\frac{13\!\cdots\!97}{45\!\cdots\!33}a^{3}+\frac{92\!\cdots\!31}{45\!\cdots\!33}a^{2}-\frac{27\!\cdots\!86}{45\!\cdots\!33}a+\frac{129422966213466}{351701785866241}$, $\frac{570164725336254}{45\!\cdots\!33}a^{24}-\frac{12\!\cdots\!76}{45\!\cdots\!33}a^{23}+\frac{35\!\cdots\!39}{45\!\cdots\!33}a^{22}-\frac{618627861648428}{351701785866241}a^{21}+\frac{946756247111964}{351701785866241}a^{20}-\frac{13\!\cdots\!88}{45\!\cdots\!33}a^{19}+\frac{76\!\cdots\!05}{45\!\cdots\!33}a^{18}+\frac{46\!\cdots\!77}{45\!\cdots\!33}a^{17}-\frac{16\!\cdots\!73}{45\!\cdots\!33}a^{16}+\frac{23\!\cdots\!76}{45\!\cdots\!33}a^{15}-\frac{16\!\cdots\!08}{45\!\cdots\!33}a^{14}-\frac{15\!\cdots\!19}{45\!\cdots\!33}a^{13}+\frac{10\!\cdots\!95}{111515200396613}a^{12}-\frac{66\!\cdots\!75}{45\!\cdots\!33}a^{11}+\frac{77\!\cdots\!55}{45\!\cdots\!33}a^{10}-\frac{42\!\cdots\!22}{45\!\cdots\!33}a^{9}+\frac{15\!\cdots\!16}{45\!\cdots\!33}a^{8}+\frac{18\!\cdots\!70}{351701785866241}a^{7}+\frac{80\!\cdots\!59}{45\!\cdots\!33}a^{6}-\frac{18\!\cdots\!56}{45\!\cdots\!33}a^{5}+\frac{29\!\cdots\!99}{45\!\cdots\!33}a^{4}+\frac{698132508869109}{45\!\cdots\!33}a^{3}+\frac{20\!\cdots\!57}{45\!\cdots\!33}a^{2}+\frac{14\!\cdots\!53}{45\!\cdots\!33}a+\frac{29\!\cdots\!21}{45\!\cdots\!33}$, $\frac{13\!\cdots\!03}{45\!\cdots\!33}a^{24}-\frac{35\!\cdots\!57}{45\!\cdots\!33}a^{23}+\frac{10\!\cdots\!63}{45\!\cdots\!33}a^{22}-\frac{634253016881830}{111515200396613}a^{21}+\frac{46\!\cdots\!46}{45\!\cdots\!33}a^{20}-\frac{66\!\cdots\!63}{45\!\cdots\!33}a^{19}+\frac{52\!\cdots\!07}{351701785866241}a^{18}-\frac{43\!\cdots\!95}{45\!\cdots\!33}a^{17}-\frac{25\!\cdots\!81}{45\!\cdots\!33}a^{16}+\frac{61\!\cdots\!13}{45\!\cdots\!33}a^{15}-\frac{88\!\cdots\!29}{45\!\cdots\!33}a^{14}+\frac{75\!\cdots\!32}{45\!\cdots\!33}a^{13}+\frac{33\!\cdots\!44}{45\!\cdots\!33}a^{12}-\frac{16\!\cdots\!84}{45\!\cdots\!33}a^{11}+\frac{31\!\cdots\!63}{45\!\cdots\!33}a^{10}-\frac{35\!\cdots\!65}{45\!\cdots\!33}a^{9}+\frac{31\!\cdots\!11}{45\!\cdots\!33}a^{8}-\frac{19\!\cdots\!48}{45\!\cdots\!33}a^{7}+\frac{14\!\cdots\!42}{45\!\cdots\!33}a^{6}-\frac{11\!\cdots\!07}{45\!\cdots\!33}a^{5}+\frac{13\!\cdots\!37}{45\!\cdots\!33}a^{4}-\frac{10\!\cdots\!15}{45\!\cdots\!33}a^{3}+\frac{63\!\cdots\!20}{45\!\cdots\!33}a^{2}-\frac{16\!\cdots\!67}{45\!\cdots\!33}a+\frac{37\!\cdots\!11}{45\!\cdots\!33}$, $\frac{22627989600720}{45\!\cdots\!33}a^{24}-\frac{288349015149901}{45\!\cdots\!33}a^{23}+\frac{379528533421111}{45\!\cdots\!33}a^{22}-\frac{15\!\cdots\!80}{45\!\cdots\!33}a^{21}+\frac{31\!\cdots\!62}{45\!\cdots\!33}a^{20}-\frac{40\!\cdots\!02}{45\!\cdots\!33}a^{19}+\frac{59\!\cdots\!89}{45\!\cdots\!33}a^{18}-\frac{57\!\cdots\!89}{45\!\cdots\!33}a^{17}+\frac{35\!\cdots\!34}{45\!\cdots\!33}a^{16}-\frac{774061446371632}{45\!\cdots\!33}a^{15}-\frac{458611130415150}{351701785866241}a^{14}+\frac{65\!\cdots\!18}{45\!\cdots\!33}a^{13}-\frac{57\!\cdots\!86}{45\!\cdots\!33}a^{12}-\frac{64\!\cdots\!85}{45\!\cdots\!33}a^{11}+\frac{12\!\cdots\!77}{45\!\cdots\!33}a^{10}-\frac{29\!\cdots\!65}{45\!\cdots\!33}a^{9}+\frac{34\!\cdots\!83}{45\!\cdots\!33}a^{8}-\frac{34\!\cdots\!33}{45\!\cdots\!33}a^{7}+\frac{22\!\cdots\!91}{45\!\cdots\!33}a^{6}-\frac{18\!\cdots\!96}{45\!\cdots\!33}a^{5}+\frac{35\!\cdots\!56}{45\!\cdots\!33}a^{4}-\frac{20\!\cdots\!41}{45\!\cdots\!33}a^{3}+\frac{550043426009773}{147487845685843}a^{2}-\frac{95\!\cdots\!57}{45\!\cdots\!33}a+\frac{52\!\cdots\!85}{45\!\cdots\!33}$, $\frac{239371565914805}{45\!\cdots\!33}a^{24}-\frac{11\!\cdots\!19}{45\!\cdots\!33}a^{23}+\frac{36\!\cdots\!01}{45\!\cdots\!33}a^{22}-\frac{91\!\cdots\!49}{45\!\cdots\!33}a^{21}+\frac{19\!\cdots\!83}{45\!\cdots\!33}a^{20}-\frac{33\!\cdots\!62}{45\!\cdots\!33}a^{19}+\frac{42\!\cdots\!06}{45\!\cdots\!33}a^{18}-\frac{39\!\cdots\!64}{45\!\cdots\!33}a^{17}+\frac{18\!\cdots\!68}{45\!\cdots\!33}a^{16}+\frac{502629880917193}{147487845685843}a^{15}-\frac{46\!\cdots\!85}{45\!\cdots\!33}a^{14}+\frac{59\!\cdots\!20}{45\!\cdots\!33}a^{13}-\frac{27\!\cdots\!27}{45\!\cdots\!33}a^{12}-\frac{33\!\cdots\!83}{351701785866241}a^{11}+\frac{14\!\cdots\!19}{45\!\cdots\!33}a^{10}-\frac{21\!\cdots\!83}{45\!\cdots\!33}a^{9}+\frac{22\!\cdots\!38}{45\!\cdots\!33}a^{8}-\frac{16\!\cdots\!96}{45\!\cdots\!33}a^{7}+\frac{98\!\cdots\!55}{45\!\cdots\!33}a^{6}-\frac{61\!\cdots\!49}{45\!\cdots\!33}a^{5}+\frac{68\!\cdots\!70}{45\!\cdots\!33}a^{4}-\frac{74\!\cdots\!19}{45\!\cdots\!33}a^{3}+\frac{58\!\cdots\!84}{45\!\cdots\!33}a^{2}-\frac{30\!\cdots\!89}{45\!\cdots\!33}a+\frac{19\!\cdots\!32}{45\!\cdots\!33}$, $\frac{684368870242743}{45\!\cdots\!33}a^{24}-\frac{15\!\cdots\!36}{45\!\cdots\!33}a^{23}+\frac{358770754405718}{351701785866241}a^{22}-\frac{10\!\cdots\!97}{45\!\cdots\!33}a^{21}+\frac{17\!\cdots\!36}{45\!\cdots\!33}a^{20}-\frac{23\!\cdots\!09}{45\!\cdots\!33}a^{19}+\frac{21\!\cdots\!37}{45\!\cdots\!33}a^{18}-\frac{96\!\cdots\!65}{45\!\cdots\!33}a^{17}-\frac{461345246825832}{351701785866241}a^{16}+\frac{25\!\cdots\!32}{45\!\cdots\!33}a^{15}-\frac{28\!\cdots\!57}{45\!\cdots\!33}a^{14}+\frac{18\!\cdots\!65}{45\!\cdots\!33}a^{13}+\frac{28\!\cdots\!62}{45\!\cdots\!33}a^{12}-\frac{71\!\cdots\!74}{45\!\cdots\!33}a^{11}+\frac{11\!\cdots\!32}{45\!\cdots\!33}a^{10}-\frac{11\!\cdots\!17}{45\!\cdots\!33}a^{9}+\frac{92\!\cdots\!23}{45\!\cdots\!33}a^{8}-\frac{51\!\cdots\!71}{45\!\cdots\!33}a^{7}+\frac{51\!\cdots\!03}{45\!\cdots\!33}a^{6}-\frac{46\!\cdots\!83}{45\!\cdots\!33}a^{5}+\frac{44\!\cdots\!88}{45\!\cdots\!33}a^{4}-\frac{30\!\cdots\!90}{45\!\cdots\!33}a^{3}+\frac{17\!\cdots\!86}{45\!\cdots\!33}a^{2}-\frac{84\!\cdots\!80}{45\!\cdots\!33}a+\frac{58\!\cdots\!59}{45\!\cdots\!33}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 593552.1381003528 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 593552.1381003528 \cdot 1}{2\cdot\sqrt{145890213878661931676924574560641}}\cr\approx \mathstrut & 0.186039093014903 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.229441.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $25$ $25$ $25$ $25$ $25$ ${\href{/padicField/13.2.0.1}{2} }^{12}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{12}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $25$ ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{12}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{12}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(479\) Copy content Toggle raw display $\Q_{479}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.479.2t1.a.a$1$ $ 479 $ \(\Q(\sqrt{-479}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.479.5t2.a.a$2$ $ 479 $ 5.1.229441.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.479.5t2.a.b$2$ $ 479 $ 5.1.229441.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.479.25t4.a.f$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.d$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.h$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.c$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.i$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.g$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.a$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.j$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.e$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.b$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.