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Properties

Label 40.0.23566017627...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 5^{70}\cdot 7^{20}$
Root discriminant $76.61$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95367431640625, 0, 0, 0, 0, 4608154296875, 0, 0, 0, 0, 253183593750, 0, 0, 0, 0, 13708440625, 0, 0, 0, 0, 743410601, 0, 0, 0, 0, -4386701, 0, 0, 0, 0, 25926, 0, 0, 0, 0, -151, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 151*x^35 + 25926*x^30 - 4386701*x^25 + 743410601*x^20 + 13708440625*x^15 + 253183593750*x^10 + 4608154296875*x^5 + 95367431640625)
 
gp: K = bnfinit(x^40 - 151*x^35 + 25926*x^30 - 4386701*x^25 + 743410601*x^20 + 13708440625*x^15 + 253183593750*x^10 + 4608154296875*x^5 + 95367431640625, 1)
 

Normalized defining polynomial

\( x^{40} - 151 x^{35} + 25926 x^{30} - 4386701 x^{25} + 743410601 x^{20} + 13708440625 x^{15} + 253183593750 x^{10} + 4608154296875 x^{5} + 95367431640625 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2356601762749139913517942415268160462338276062155273393727838993072509765625=3^{20}\cdot 5^{70}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.61$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(525=3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{525}(1,·)$, $\chi_{525}(524,·)$, $\chi_{525}(398,·)$, $\chi_{525}(272,·)$, $\chi_{525}(274,·)$, $\chi_{525}(148,·)$, $\chi_{525}(22,·)$, $\chi_{525}(419,·)$, $\chi_{525}(293,·)$, $\chi_{525}(167,·)$, $\chi_{525}(41,·)$, $\chi_{525}(43,·)$, $\chi_{525}(169,·)$, $\chi_{525}(314,·)$, $\chi_{525}(316,·)$, $\chi_{525}(62,·)$, $\chi_{525}(64,·)$, $\chi_{525}(461,·)$, $\chi_{525}(463,·)$, $\chi_{525}(337,·)$, $\chi_{525}(83,·)$, $\chi_{525}(356,·)$, $\chi_{525}(442,·)$, $\chi_{525}(421,·)$, $\chi_{525}(482,·)$, $\chi_{525}(379,·)$, $\chi_{525}(484,·)$, $\chi_{525}(358,·)$, $\chi_{525}(209,·)$, $\chi_{525}(104,·)$, $\chi_{525}(188,·)$, $\chi_{525}(106,·)$, $\chi_{525}(146,·)$, $\chi_{525}(232,·)$, $\chi_{525}(211,·)$, $\chi_{525}(503,·)$, $\chi_{525}(377,·)$, $\chi_{525}(251,·)$, $\chi_{525}(253,·)$, $\chi_{525}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{41} a^{20} - \frac{14}{41} a^{15} - \frac{9}{41} a^{10} + \frac{3}{41} a^{5} - \frac{1}{41}$, $\frac{1}{205} a^{21} - \frac{96}{205} a^{16} - \frac{9}{205} a^{11} + \frac{44}{205} a^{6} + \frac{81}{205} a$, $\frac{1}{1025} a^{22} - \frac{301}{1025} a^{17} + \frac{401}{1025} a^{12} + \frac{249}{1025} a^{7} - \frac{124}{1025} a^{2}$, $\frac{1}{5125} a^{23} + \frac{724}{5125} a^{18} + \frac{1426}{5125} a^{13} + \frac{2299}{5125} a^{8} - \frac{1149}{5125} a^{3}$, $\frac{1}{25625} a^{24} - \frac{9526}{25625} a^{19} + \frac{6551}{25625} a^{14} - \frac{7951}{25625} a^{9} - \frac{6274}{25625} a^{4}$, $\frac{1}{95249483253125} a^{25} - \frac{176}{128125} a^{20} + \frac{10951}{128125} a^{15} - \frac{25476}{128125} a^{10} + \frac{15626}{128125} a^{5} + \frac{8925313913}{30479834641}$, $\frac{1}{476247416265625} a^{26} - \frac{176}{640625} a^{21} + \frac{139076}{640625} a^{16} + \frac{230774}{640625} a^{11} + \frac{15626}{640625} a^{6} - \frac{52034355369}{152399173205} a$, $\frac{1}{2381237081328125} a^{27} - \frac{176}{3203125} a^{22} - \frac{1142174}{3203125} a^{17} - \frac{1050476}{3203125} a^{12} + \frac{656251}{3203125} a^{7} + \frac{252763991041}{761995866025} a^{2}$, $\frac{1}{11906185406640625} a^{28} - \frac{176}{16015625} a^{23} + \frac{2060951}{16015625} a^{18} - \frac{7456726}{16015625} a^{13} + \frac{7062501}{16015625} a^{8} + \frac{1014759857066}{3809979330125} a^{3}$, $\frac{1}{59530927033203125} a^{29} - \frac{176}{80078125} a^{24} + \frac{2060951}{80078125} a^{19} + \frac{8558899}{80078125} a^{14} + \frac{23078126}{80078125} a^{9} - \frac{6605198803184}{19049896650625} a^{4}$, $\frac{1}{297654635166015625} a^{30} - \frac{151}{297654635166015625} a^{25} + \frac{2029701}{400390625} a^{20} - \frac{14066101}{400390625} a^{15} + \frac{58593751}{400390625} a^{10} - \frac{4646311869549}{95249483253125} a^{5} - \frac{9664311887}{30479834641}$, $\frac{1}{1488273175830078125} a^{31} - \frac{151}{1488273175830078125} a^{26} + \frac{2029701}{2001953125} a^{21} - \frac{14066101}{2001953125} a^{16} + \frac{458984376}{2001953125} a^{11} - \frac{4646311869549}{476247416265625} a^{6} - \frac{9664311887}{152399173205} a$, $\frac{1}{7441365879150390625} a^{32} - \frac{151}{7441365879150390625} a^{27} + \frac{2029701}{10009765625} a^{22} + \frac{1987887024}{10009765625} a^{17} - \frac{3544921874}{10009765625} a^{12} - \frac{957141144400799}{2381237081328125} a^{7} - \frac{9664311887}{761995866025} a^{2}$, $\frac{1}{37206829395751953125} a^{33} - \frac{151}{37206829395751953125} a^{28} + \frac{2029701}{50048828125} a^{23} + \frac{11997652649}{50048828125} a^{18} - \frac{3544921874}{50048828125} a^{13} - \frac{3338378225728924}{11906185406640625} a^{8} - \frac{9664311887}{3809979330125} a^{3}$, $\frac{1}{186034146978759765625} a^{34} - \frac{151}{186034146978759765625} a^{29} + \frac{2029701}{250244140625} a^{24} - \frac{88100003601}{250244140625} a^{19} + \frac{46503906251}{250244140625} a^{14} - \frac{15244563632369549}{59530927033203125} a^{9} + \frac{7610294348363}{19049896650625} a^{4}$, $\frac{1}{930170734893798828125} a^{35} - \frac{151}{930170734893798828125} a^{30} + \frac{25926}{930170734893798828125} a^{25} + \frac{1860933899}{1251220703125} a^{20} - \frac{335693359374}{1251220703125} a^{15} - \frac{72598691499519549}{297654635166015625} a^{10} + \frac{39493688204051}{95249483253125} a^{5} + \frac{5947284959}{30479834641}$, $\frac{1}{4650853674468994140625} a^{36} - \frac{151}{4650853674468994140625} a^{31} + \frac{25926}{4650853674468994140625} a^{26} + \frac{1860933899}{6256103515625} a^{21} - \frac{335693359374}{6256103515625} a^{16} + \frac{225055943666496076}{1488273175830078125} a^{11} + \frac{229992654710301}{476247416265625} a^{6} - \frac{55012384323}{152399173205} a$, $\frac{1}{23254268372344970703125} a^{37} - \frac{151}{23254268372344970703125} a^{32} + \frac{25926}{23254268372344970703125} a^{27} + \frac{1860933899}{31280517578125} a^{22} - \frac{12847900390624}{31280517578125} a^{17} - \frac{2751490407993660174}{7441365879150390625} a^{12} + \frac{706240070975926}{2381237081328125} a^{7} - \frac{207411557528}{761995866025} a^{2}$, $\frac{1}{116271341861724853515625} a^{38} - \frac{151}{116271341861724853515625} a^{33} + \frac{25926}{116271341861724853515625} a^{28} + \frac{1860933899}{156402587890625} a^{23} - \frac{12847900390624}{156402587890625} a^{18} - \frac{17634222166294441424}{37206829395751953125} a^{13} - \frac{1674997010352199}{11906185406640625} a^{8} + \frac{554584308497}{3809979330125} a^{3}$, $\frac{1}{581356709308624267578125} a^{39} - \frac{151}{581356709308624267578125} a^{34} + \frac{25926}{581356709308624267578125} a^{29} + \frac{1860933899}{782012939453125} a^{24} - \frac{169250488281249}{782012939453125} a^{19} - \frac{54841051562046394549}{186034146978759765625} a^{14} + \frac{10231188396288426}{59530927033203125} a^{9} - \frac{3255395021628}{19049896650625} a^{4}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{6}{3809979330125} a^{28} - \frac{662854323131}{3809979330125} a^{3} \) (order $50$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 4.0.55125.1, \(\Q(\zeta_{5})\), 5.5.390625.1, 8.0.3038765625.3, 10.10.3115921783447265625.1, \(\Q(\zeta_{25})^+\), 10.10.623184356689453125.1, 20.20.9708968560561188496649265289306640625.1, 20.0.48544842802805942483246326446533203125.1, \(\Q(\zeta_{25})\)

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 $20^{2}$ R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$