LMFDB - Number field 25.1.34694469519536141888238489627838134765625.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32)

gp: K = bnfinit(y^25 - 135*y^20 - 460*y^15 - 1580*y^10 + 455*y^5 - 32, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32)

$ x^{25} - 135x^{20} - 460x^{15} - 1580x^{10} + 455x^{5} - 32 $ x^25 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32

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:	$34694469519536141888238489627838134765625$ 34694469519536141888238489627838134765625 $\medspace = 5^{58}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$41.84$	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:	$5^{239/100}\approx 46.83152609852333$
Ramified primes:	$5$ 5	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{5}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{6}-\frac{1}{5}a$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{25}a^{13}-\frac{1}{25}a^{12}+\frac{2}{25}a^{11}+\frac{2}{25}a^{10}+\frac{1}{25}a^{8}-\frac{1}{25}a^{7}+\frac{2}{25}a^{6}+\frac{2}{25}a^{5}-\frac{6}{25}a^{3}+\frac{6}{25}a^{2}-\frac{12}{25}a-\frac{12}{25}$, $\frac{1}{25}a^{14}+\frac{1}{25}a^{12}-\frac{1}{25}a^{11}+\frac{2}{25}a^{10}+\frac{1}{25}a^{9}+\frac{1}{25}a^{7}-\frac{1}{25}a^{6}+\frac{2}{25}a^{5}-\frac{6}{25}a^{4}-\frac{6}{25}a^{2}+\frac{6}{25}a-\frac{12}{25}$, $\frac{1}{25}a^{15}-\frac{1}{25}a^{10}-\frac{8}{25}a^{5}+\frac{12}{25}$, $\frac{1}{25}a^{16}-\frac{1}{25}a^{11}-\frac{8}{25}a^{6}+\frac{12}{25}a$, $\frac{1}{25}a^{17}-\frac{1}{25}a^{12}+\frac{2}{25}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{8}{25}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{25}a^{18}-\frac{1}{25}a^{12}+\frac{2}{25}a^{11}+\frac{2}{25}a^{10}-\frac{2}{25}a^{8}-\frac{1}{25}a^{7}-\frac{8}{25}a^{6}+\frac{7}{25}a^{5}-\frac{4}{25}a^{3}+\frac{6}{25}a^{2}+\frac{8}{25}a+\frac{3}{25}$, $\frac{1}{125}a^{19}+\frac{2}{125}a^{18}-\frac{1}{125}a^{17}-\frac{2}{125}a^{16}+\frac{1}{125}a^{15}-\frac{1}{125}a^{14}-\frac{2}{125}a^{13}+\frac{1}{125}a^{12}+\frac{2}{125}a^{11}-\frac{1}{125}a^{10}-\frac{8}{125}a^{9}+\frac{9}{125}a^{8}+\frac{8}{125}a^{7}-\frac{34}{125}a^{6}+\frac{17}{125}a^{5}+\frac{12}{125}a^{4}-\frac{26}{125}a^{3}-\frac{12}{125}a^{2}-\frac{49}{125}a-\frac{38}{125}$, $\frac{1}{75125}a^{20}+\frac{1382}{75125}a^{15}-\frac{7466}{75125}a^{10}+\frac{16373}{75125}a^{5}-\frac{28079}{75125}$, $\frac{1}{150250}a^{21}-\frac{1623}{150250}a^{16}+\frac{5282}{75125}a^{11}+\frac{27719}{75125}a^{6}-\frac{4039}{150250}a$, $\frac{1}{300500}a^{22}-\frac{1623}{300500}a^{17}+\frac{2641}{75125}a^{12}+\frac{6347}{75125}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{94189}{300500}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{601000}a^{23}-\frac{1623}{601000}a^{18}+\frac{2641}{150250}a^{13}+\frac{6347}{150250}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{206311}{601000}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{1202000}a^{24}-\frac{1623}{1202000}a^{19}+\frac{2641}{300500}a^{14}-\frac{23703}{300500}a^{9}+\frac{446711}{1202000}a^{4}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/5*a^7 + 1/5*a^6 - 1/5*a^5 - 2/5*a^2 - 2/5*a + 2/5, 1/5*a^8 - 2/5*a^6 + 1/5*a^5 - 2/5*a^3 - 1/5*a - 2/5, 1/5*a^9 - 2/5*a^6 - 2/5*a^5 - 2/5*a^4 - 1/5*a - 1/5, 1/5*a^10 + 1/5*a^5 - 1/5, 1/5*a^11 + 1/5*a^6 - 1/5*a, 1/5*a^12 - 1/5*a^6 + 1/5*a^5 + 1/5*a^2 + 2/5*a - 2/5, 1/25*a^13 - 1/25*a^12 + 2/25*a^11 + 2/25*a^10 + 1/25*a^8 - 1/25*a^7 + 2/25*a^6 + 2/25*a^5 - 6/25*a^3 + 6/25*a^2 - 12/25*a - 12/25, 1/25*a^14 + 1/25*a^12 - 1/25*a^11 + 2/25*a^10 + 1/25*a^9 + 1/25*a^7 - 1/25*a^6 + 2/25*a^5 - 6/25*a^4 - 6/25*a^2 + 6/25*a - 12/25, 1/25*a^15 - 1/25*a^10 - 8/25*a^5 + 12/25, 1/25*a^16 - 1/25*a^11 - 8/25*a^6 + 12/25*a, 1/25*a^17 - 1/25*a^12 + 2/25*a^7 + 2/5*a^6 - 2/5*a^5 - 8/25*a^2 + 1/5*a - 1/5, 1/25*a^18 - 1/25*a^12 + 2/25*a^11 + 2/25*a^10 - 2/25*a^8 - 1/25*a^7 - 8/25*a^6 + 7/25*a^5 - 4/25*a^3 + 6/25*a^2 + 8/25*a + 3/25, 1/125*a^19 + 2/125*a^18 - 1/125*a^17 - 2/125*a^16 + 1/125*a^15 - 1/125*a^14 - 2/125*a^13 + 1/125*a^12 + 2/125*a^11 - 1/125*a^10 - 8/125*a^9 + 9/125*a^8 + 8/125*a^7 - 34/125*a^6 + 17/125*a^5 + 12/125*a^4 - 26/125*a^3 - 12/125*a^2 - 49/125*a - 38/125, 1/75125*a^20 + 1382/75125*a^15 - 7466/75125*a^10 + 16373/75125*a^5 - 28079/75125, 1/150250*a^21 - 1623/150250*a^16 + 5282/75125*a^11 + 27719/75125*a^6 - 4039/150250*a, 1/300500*a^22 - 1623/300500*a^17 + 2641/75125*a^12 + 6347/75125*a^7 + 2/5*a^6 - 2/5*a^5 - 94189/300500*a^2 + 1/5*a - 1/5, 1/601000*a^23 - 1623/601000*a^18 + 2641/150250*a^13 + 6347/150250*a^8 - 2/5*a^6 + 1/5*a^5 + 206311/601000*a^3 - 1/5*a - 2/5, 1/1202000*a^24 - 1623/1202000*a^19 + 2641/300500*a^14 - 23703/300500*a^9 + 446711/1202000*a^4

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 $ -1 (order $2$)	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:	$\frac{55087}{1202000}a^{24}-\frac{7429801}{1202000}a^{19}-\frac{6568233}{300500}a^{14}-\frac{22712361}{300500}a^{9}+\frac{12885257}{1202000}a^{4}$, $\frac{22713}{1202000}a^{24}-\frac{7269}{601000}a^{23}+\frac{5321}{75125}a^{22}+\frac{11603}{150250}a^{21}+\frac{481}{15025}a^{20}+\frac{3062959}{1202000}a^{19}+\frac{979587}{601000}a^{18}-\frac{717808}{75125}a^{17}-\frac{1564939}{150250}a^{16}-\frac{64872}{15025}a^{15}+\frac{2723307}{300500}a^{14}+\frac{894001}{150250}a^{13}-\frac{2518676}{75125}a^{12}-\frac{2767534}{75125}a^{11}-\frac{229753}{15025}a^{10}+\frac{9341999}{300500}a^{9}+\frac{3103587}{150250}a^{8}-\frac{8666647}{75125}a^{7}-\frac{9518428}{75125}a^{6}-\frac{790406}{15025}a^{5}-\frac{5834543}{1202000}a^{4}+\frac{229021}{601000}a^{3}+\frac{1478701}{75125}a^{2}+\frac{2862223}{150250}a+\frac{108454}{15025}$, $\frac{81643}{601000}a^{24}-\frac{23601}{300500}a^{22}-\frac{5771}{150250}a^{21}+\frac{1672}{75125}a^{20}-\frac{11008429}{601000}a^{19}+\frac{3181983}{300500}a^{17}+\frac{778043}{150250}a^{16}-\frac{225516}{75125}a^{15}-\frac{9839977}{150250}a^{14}+\frac{2854219}{75125}a^{12}+\frac{1397623}{75125}a^{11}-\frac{796707}{75125}a^{10}-\frac{33842399}{150250}a^{9}+\frac{9803008}{75125}a^{7}+\frac{4803811}{75125}a^{6}-\frac{2728434}{75125}a^{5}+\frac{15320093}{601000}a^{4}-\frac{3918691}{300500}a^{2}-\frac{839111}{150250}a+\frac{401697}{75125}$, $\frac{2759}{240400}a^{24}-\frac{34441}{601000}a^{23}+\frac{1441}{60100}a^{22}+\frac{3344}{75125}a^{21}-\frac{14873}{9616}a^{19}+\frac{4644463}{601000}a^{18}-\frac{7775}{2404}a^{17}-\frac{451032}{75125}a^{16}-\frac{338969}{60100}a^{14}+\frac{4131539}{150250}a^{13}-\frac{171136}{15025}a^{12}-\frac{1593414}{75125}a^{11}-\frac{1152081}{60100}a^{9}+\frac{14236383}{150250}a^{8}-\frac{584764}{15025}a^{7}-\frac{5456868}{75125}a^{6}+\frac{416033}{240400}a^{4}-\frac{7482711}{601000}a^{3}+\frac{433807}{60100}a^{2}+\frac{728269}{75125}a$, $\frac{12808}{75125}a^{24}+\frac{43051}{300500}a^{23}+\frac{47939}{300500}a^{22}-\frac{861}{30050}a^{21}-\frac{8687}{75125}a^{20}-\frac{1726669}{75125}a^{19}-\frac{5805173}{300500}a^{18}-\frac{6466297}{300500}a^{17}+\frac{116071}{30050}a^{16}+\frac{1171491}{75125}a^{15}-\frac{6216763}{75125}a^{14}-\frac{5177104}{75125}a^{13}-\frac{5697241}{75125}a^{12}+\frac{209113}{15025}a^{11}+\frac{4165157}{75125}a^{10}-\frac{21400151}{75125}a^{9}-\frac{17816598}{75125}a^{8}-\frac{19604057}{75125}a^{7}+\frac{716043}{15025}a^{6}+\frac{14322239}{75125}a^{5}+\frac{1856653}{75125}a^{4}+\frac{8598641}{300500}a^{3}+\frac{12763789}{300500}a^{2}-\frac{141643}{30050}a-\frac{2000917}{75125}$, $\frac{23451}{75125}a^{24}-\frac{74343}{120200}a^{23}-\frac{21984}{75125}a^{22}+\frac{5321}{75125}a^{21}+\frac{1403}{15025}a^{20}+\frac{3163908}{75125}a^{19}+\frac{10026609}{120200}a^{18}+\frac{2964597}{75125}a^{17}-\frac{717808}{75125}a^{16}-\frac{37839}{3005}a^{15}+\frac{11054236}{75125}a^{14}+\frac{1775279}{6010}a^{13}+\frac{10549999}{75125}a^{12}-\frac{2518676}{75125}a^{11}-\frac{134738}{3005}a^{10}+\frac{37977842}{75125}a^{9}+\frac{30519677}{30050}a^{8}+\frac{36286253}{75125}a^{7}-\frac{8666647}{75125}a^{6}-\frac{2318761}{15025}a^{5}-\frac{7460426}{75125}a^{4}-\frac{3594269}{24040}a^{3}-\frac{4608034}{75125}a^{2}+\frac{1478701}{75125}a+\frac{289794}{15025}$, $\frac{254773}{601000}a^{24}-\frac{91419}{601000}a^{23}-\frac{1403}{15025}a^{22}+\frac{961}{150250}a^{21}-\frac{3138}{75125}a^{20}-\frac{34361739}{601000}a^{19}+\frac{12330677}{601000}a^{18}+\frac{37839}{3005}a^{17}-\frac{129323}{150250}a^{16}+\frac{423204}{75125}a^{15}-\frac{30398647}{150250}a^{14}+\frac{10880251}{150250}a^{13}+\frac{134738}{3005}a^{12}-\frac{248858}{75125}a^{11}+\frac{1500823}{75125}a^{10}-\frac{104526279}{150250}a^{9}+\frac{37417317}{150250}a^{8}+\frac{2318761}{15025}a^{7}-\frac{851781}{75125}a^{6}+\frac{5175626}{75125}a^{5}+\frac{62478003}{601000}a^{4}-\frac{23650389}{601000}a^{3}-\frac{289794}{15025}a^{2}-\frac{245429}{150250}a-\frac{682843}{75125}$, $\frac{254773}{601000}a^{24}-\frac{51661}{601000}a^{23}-\frac{47939}{300500}a^{22}-\frac{58}{3005}a^{21}+\frac{1403}{15025}a^{20}-\frac{34361739}{601000}a^{19}+\frac{6965883}{601000}a^{18}+\frac{6466297}{300500}a^{17}+\frac{39152}{15025}a^{16}-\frac{37839}{3005}a^{15}-\frac{30398647}{150250}a^{14}+\frac{6222669}{150250}a^{13}+\frac{5697241}{75125}a^{12}+\frac{133159}{15025}a^{11}-\frac{134738}{3005}a^{10}-\frac{104526279}{150250}a^{9}+\frac{21396813}{150250}a^{8}+\frac{19604057}{75125}a^{7}+\frac{90697}{3005}a^{6}-\frac{2318761}{15025}a^{5}+\frac{62478003}{601000}a^{4}-\frac{9714571}{601000}a^{3}-\frac{12763789}{300500}a^{2}-\frac{161107}{15025}a+\frac{304819}{15025}$, $\frac{55573}{1202000}a^{24}+\frac{13069}{300500}a^{23}-\frac{10137}{300500}a^{22}-\frac{3449}{150250}a^{21}-\frac{422}{75125}a^{20}-\frac{7497379}{1202000}a^{19}-\frac{1762627}{300500}a^{18}+\frac{1367251}{300500}a^{17}+\frac{465187}{150250}a^{16}+\frac{56861}{75125}a^{15}-\frac{6558827}{300500}a^{14}-\frac{1559796}{75125}a^{13}+\frac{1207738}{75125}a^{12}+\frac{822062}{75125}a^{11}+\frac{208757}{75125}a^{10}-\frac{22524539}{300500}a^{9}-\frac{5371012}{75125}a^{8}+\frac{4147416}{75125}a^{7}+\frac{2837964}{75125}a^{6}+\frac{723294}{75125}a^{5}+\frac{17521283}{1202000}a^{4}+\frac{2993119}{300500}a^{3}-\frac{2731327}{300500}a^{2}-\frac{824039}{150250}a-\frac{74502}{75125}$, $\frac{193951}{601000}a^{23}-\frac{26158233}{601000}a^{18}-\frac{23152219}{150250}a^{13}-\frac{79641783}{150250}a^{8}+\frac{46691041}{601000}a^{3}$, $\frac{632}{15025}a^{24}-\frac{7269}{601000}a^{23}+\frac{939}{75125}a^{22}+\frac{1063}{30050}a^{21}-\frac{2611}{75125}a^{20}-\frac{85171}{15025}a^{19}+\frac{979587}{601000}a^{18}-\frac{126672}{75125}a^{17}-\frac{143417}{30050}a^{16}+\frac{352188}{75125}a^{15}-\frac{310778}{15025}a^{14}+\frac{894001}{150250}a^{13}-\frac{444649}{75125}a^{12}-\frac{250393}{15025}a^{11}+\frac{1241356}{75125}a^{10}-\frac{214974}{3005}a^{9}+\frac{3103587}{150250}a^{8}-\frac{1504838}{75125}a^{7}-\frac{864769}{15025}a^{6}+\frac{4233272}{75125}a^{5}+\frac{35258}{15025}a^{4}+\frac{229021}{601000}a^{3}+\frac{261124}{75125}a^{2}+\frac{292173}{30050}a-\frac{662821}{75125}$, $\frac{28531}{150250}a^{24}-\frac{159047}{300500}a^{23}+\frac{66213}{150250}a^{22}-\frac{20068}{75125}a^{21}+\frac{4899}{75125}a^{20}-\frac{3845163}{150250}a^{19}+\frac{21447461}{300500}a^{18}-\frac{8929749}{150250}a^{17}+\frac{2706674}{75125}a^{16}-\frac{660947}{75125}a^{15}-\frac{7001658}{75125}a^{14}+\frac{19095553}{75125}a^{13}-\frac{15836209}{75125}a^{12}+\frac{9569313}{75125}a^{11}-\frac{2309919}{75125}a^{10}-\frac{24135261}{75125}a^{9}+\frac{65696656}{75125}a^{8}-\frac{54471328}{75125}a^{7}+\frac{32898086}{75125}a^{6}-\frac{7943353}{75125}a^{5}+\frac{2078491}{150250}a^{4}-\frac{32826657}{300500}a^{3}+\frac{15185943}{150250}a^{2}-\frac{5041603}{75125}a+\frac{1554449}{75125}$ 55087/1202000a^24 - 7429801/1202000a^19 - 6568233/300500a^14 - 22712361/300500a^9 + 12885257/1202000a^4, -22713/1202000a^24 - 7269/601000a^23 + 5321/75125a^22 + 11603/150250a^21 + 481/15025a^20 + 3062959/1202000a^19 + 979587/601000a^18 - 717808/75125a^17 - 1564939/150250a^16 - 64872/15025a^15 + 2723307/300500a^14 + 894001/150250a^13 - 2518676/75125a^12 - 2767534/75125a^11 - 229753/15025a^10 + 9341999/300500a^9 + 3103587/150250a^8 - 8666647/75125a^7 - 9518428/75125a^6 - 790406/15025a^5 - 5834543/1202000a^4 + 229021/601000a^3 + 1478701/75125a^2 + 2862223/150250a + 108454/15025, 81643/601000a^24 - 23601/300500a^22 - 5771/150250a^21 + 1672/75125a^20 - 11008429/601000a^19 + 3181983/300500a^17 + 778043/150250a^16 - 225516/75125a^15 - 9839977/150250a^14 + 2854219/75125a^12 + 1397623/75125a^11 - 796707/75125a^10 - 33842399/150250a^9 + 9803008/75125a^7 + 4803811/75125a^6 - 2728434/75125a^5 + 15320093/601000a^4 - 3918691/300500a^2 - 839111/150250a + 401697/75125, 2759/240400a^24 - 34441/601000a^23 + 1441/60100a^22 + 3344/75125a^21 - 14873/9616a^19 + 4644463/601000a^18 - 7775/2404a^17 - 451032/75125a^16 - 338969/60100a^14 + 4131539/150250a^13 - 171136/15025a^12 - 1593414/75125a^11 - 1152081/60100a^9 + 14236383/150250a^8 - 584764/15025a^7 - 5456868/75125a^6 + 416033/240400a^4 - 7482711/601000a^3 + 433807/60100a^2 + 728269/75125a, 12808/75125a^24 + 43051/300500a^23 + 47939/300500a^22 - 861/30050a^21 - 8687/75125a^20 - 1726669/75125a^19 - 5805173/300500a^18 - 6466297/300500a^17 + 116071/30050a^16 + 1171491/75125a^15 - 6216763/75125a^14 - 5177104/75125a^13 - 5697241/75125a^12 + 209113/15025a^11 + 4165157/75125a^10 - 21400151/75125a^9 - 17816598/75125a^8 - 19604057/75125a^7 + 716043/15025a^6 + 14322239/75125a^5 + 1856653/75125a^4 + 8598641/300500a^3 + 12763789/300500a^2 - 141643/30050a - 2000917/75125, -23451/75125a^24 - 74343/120200a^23 - 21984/75125a^22 + 5321/75125a^21 + 1403/15025a^20 + 3163908/75125a^19 + 10026609/120200a^18 + 2964597/75125a^17 - 717808/75125a^16 - 37839/3005a^15 + 11054236/75125a^14 + 1775279/6010a^13 + 10549999/75125a^12 - 2518676/75125a^11 - 134738/3005a^10 + 37977842/75125a^9 + 30519677/30050a^8 + 36286253/75125a^7 - 8666647/75125a^6 - 2318761/15025a^5 - 7460426/75125a^4 - 3594269/24040a^3 - 4608034/75125a^2 + 1478701/75125a + 289794/15025, 254773/601000a^24 - 91419/601000a^23 - 1403/15025a^22 + 961/150250a^21 - 3138/75125a^20 - 34361739/601000a^19 + 12330677/601000a^18 + 37839/3005a^17 - 129323/150250a^16 + 423204/75125a^15 - 30398647/150250a^14 + 10880251/150250a^13 + 134738/3005a^12 - 248858/75125a^11 + 1500823/75125a^10 - 104526279/150250a^9 + 37417317/150250a^8 + 2318761/15025a^7 - 851781/75125a^6 + 5175626/75125a^5 + 62478003/601000a^4 - 23650389/601000a^3 - 289794/15025a^2 - 245429/150250a - 682843/75125, 254773/601000a^24 - 51661/601000a^23 - 47939/300500a^22 - 58/3005a^21 + 1403/15025a^20 - 34361739/601000a^19 + 6965883/601000a^18 + 6466297/300500a^17 + 39152/15025a^16 - 37839/3005a^15 - 30398647/150250a^14 + 6222669/150250a^13 + 5697241/75125a^12 + 133159/15025a^11 - 134738/3005a^10 - 104526279/150250a^9 + 21396813/150250a^8 + 19604057/75125a^7 + 90697/3005a^6 - 2318761/15025a^5 + 62478003/601000a^4 - 9714571/601000a^3 - 12763789/300500a^2 - 161107/15025a + 304819/15025, 55573/1202000a^24 + 13069/300500a^23 - 10137/300500a^22 - 3449/150250a^21 - 422/75125a^20 - 7497379/1202000a^19 - 1762627/300500a^18 + 1367251/300500a^17 + 465187/150250a^16 + 56861/75125a^15 - 6558827/300500a^14 - 1559796/75125a^13 + 1207738/75125a^12 + 822062/75125a^11 + 208757/75125a^10 - 22524539/300500a^9 - 5371012/75125a^8 + 4147416/75125a^7 + 2837964/75125a^6 + 723294/75125a^5 + 17521283/1202000a^4 + 2993119/300500a^3 - 2731327/300500a^2 - 824039/150250a - 74502/75125, 193951/601000a^23 - 26158233/601000a^18 - 23152219/150250a^13 - 79641783/150250a^8 + 46691041/601000a^3, 632/15025a^24 - 7269/601000a^23 + 939/75125a^22 + 1063/30050a^21 - 2611/75125a^20 - 85171/15025a^19 + 979587/601000a^18 - 126672/75125a^17 - 143417/30050a^16 + 352188/75125a^15 - 310778/15025a^14 + 894001/150250a^13 - 444649/75125a^12 - 250393/15025a^11 + 1241356/75125a^10 - 214974/3005a^9 + 3103587/150250a^8 - 1504838/75125a^7 - 864769/15025a^6 + 4233272/75125a^5 + 35258/15025a^4 + 229021/601000a^3 + 261124/75125a^2 + 292173/30050a - 662821/75125, 28531/150250a^24 - 159047/300500a^23 + 66213/150250a^22 - 20068/75125a^21 + 4899/75125a^20 - 3845163/150250a^19 + 21447461/300500a^18 - 8929749/150250a^17 + 2706674/75125a^16 - 660947/75125a^15 - 7001658/75125a^14 + 19095553/75125a^13 - 15836209/75125a^12 + 9569313/75125a^11 - 2309919/75125a^10 - 24135261/75125a^9 + 65696656/75125a^8 - 54471328/75125a^7 + 32898086/75125a^6 - 7943353/75125a^5 + 2078491/150250a^4 - 32826657/300500a^3 + 15185943/150250a^2 - 5041603/75125*a + 1554449/75125 (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:	$ 50169770016.52367 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 50169770016.52367 \cdot 1}{2\cdot\sqrt{34694469519536141888238489627838134765625}}\cr\approx \mathstrut & 1.01969526991187 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^25 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32)

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 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32, 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 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32);

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 - 135*x^20 - 460*x^15 - 1580*x^10 + 455*x^5 - 32);

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

$C_5^2:C_{20}$ (as 25T34):

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 500

The 26 conjugacy class representatives for $C_5^2:C_{20}$

Character table for $C_5^2:C_{20}$

Intermediate fields

5.1.1953125.1

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]

Sibling fields

Degree 25 sibling:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$	$20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$	R	${\href{/padicField/7.4.0.1}{4} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.5.0.1}{5} }^{5}$	$20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	$20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$	$20{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$	$20{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.5.0.1}{5} }^{5}$	${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	$20{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$5$ 5		Deg $25$	$25$	$1$	$58$

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