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Properties

Label 40.0.34425379066...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{70}\cdot 17^{20}$
Root discriminant $68.93$
Ramified primes $5, 17$
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![1099511627776, 0, 0, 0, 0, 108447924224, 0, 0, 0, 0, 11770265600, 0, 0, 0, 0, 1266840576, 0, 0, 0, 0, 136446449, 0, 0, 0, 0, -1237149, 0, 0, 0, 0, 11225, 0, 0, 0, 0, -101, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 101*x^35 + 11225*x^30 - 1237149*x^25 + 136446449*x^20 + 1266840576*x^15 + 11770265600*x^10 + 108447924224*x^5 + 1099511627776)
 
gp: K = bnfinit(x^40 - 101*x^35 + 11225*x^30 - 1237149*x^25 + 136446449*x^20 + 1266840576*x^15 + 11770265600*x^10 + 108447924224*x^5 + 1099511627776, 1)
 

Normalized defining polynomial

\( x^{40} - 101 x^{35} + 11225 x^{30} - 1237149 x^{25} + 136446449 x^{20} + 1266840576 x^{15} + 11770265600 x^{10} + 108447924224 x^{5} + 1099511627776 \)

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:  \(34425379066961841688086546115648602095493657770930440165102481842041015625=5^{70}\cdot 17^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.93$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(425=5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{425}(256,·)$, $\chi_{425}(1,·)$, $\chi_{425}(392,·)$, $\chi_{425}(137,·)$, $\chi_{425}(271,·)$, $\chi_{425}(16,·)$, $\chi_{425}(273,·)$, $\chi_{425}(18,·)$, $\chi_{425}(407,·)$, $\chi_{425}(152,·)$, $\chi_{425}(409,·)$, $\chi_{425}(154,·)$, $\chi_{425}(288,·)$, $\chi_{425}(33,·)$, $\chi_{425}(424,·)$, $\chi_{425}(169,·)$, $\chi_{425}(171,·)$, $\chi_{425}(307,·)$, $\chi_{425}(52,·)$, $\chi_{425}(186,·)$, $\chi_{425}(188,·)$, $\chi_{425}(322,·)$, $\chi_{425}(67,·)$, $\chi_{425}(324,·)$, $\chi_{425}(69,·)$, $\chi_{425}(203,·)$, $\chi_{425}(339,·)$, $\chi_{425}(84,·)$, $\chi_{425}(341,·)$, $\chi_{425}(86,·)$, $\chi_{425}(222,·)$, $\chi_{425}(356,·)$, $\chi_{425}(101,·)$, $\chi_{425}(358,·)$, $\chi_{425}(103,·)$, $\chi_{425}(237,·)$, $\chi_{425}(239,·)$, $\chi_{425}(373,·)$, $\chi_{425}(118,·)$, $\chi_{425}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{29} a^{20} - \frac{7}{29} a^{15} - \frac{9}{29} a^{10} + \frac{5}{29} a^{5} - \frac{6}{29}$, $\frac{1}{116} a^{21} + \frac{51}{116} a^{16} + \frac{49}{116} a^{11} - \frac{53}{116} a^{6} - \frac{35}{116} a$, $\frac{1}{464} a^{22} - \frac{181}{464} a^{17} - \frac{183}{464} a^{12} + \frac{179}{464} a^{7} + \frac{81}{464} a^{2}$, $\frac{1}{1856} a^{23} + \frac{283}{1856} a^{18} + \frac{281}{1856} a^{13} - \frac{285}{1856} a^{8} - \frac{847}{1856} a^{3}$, $\frac{1}{7424} a^{24} - \frac{3429}{7424} a^{19} - \frac{1575}{7424} a^{14} + \frac{3427}{7424} a^{9} + \frac{1009}{7424} a^{4}$, $\frac{1}{4051913749504} a^{25} - \frac{437}{29696} a^{20} + \frac{6249}{29696} a^{15} - \frac{9581}{29696} a^{10} + \frac{2049}{29696} a^{5} - \frac{1909013137}{3956947021}$, $\frac{1}{16207654998016} a^{26} - \frac{437}{118784} a^{21} - \frac{53143}{118784} a^{16} + \frac{49811}{118784} a^{11} - \frac{57343}{118784} a^{6} + \frac{6004880905}{15827788084} a$, $\frac{1}{64830619992064} a^{27} - \frac{437}{475136} a^{22} - \frac{53143}{475136} a^{17} + \frac{168595}{475136} a^{12} - \frac{176127}{475136} a^{7} + \frac{6004880905}{63311152336} a^{2}$, $\frac{1}{259322479968256} a^{28} - \frac{437}{1900544} a^{23} - \frac{528279}{1900544} a^{18} - \frac{306541}{1900544} a^{13} - \frac{651263}{1900544} a^{8} + \frac{6004880905}{253244609344} a^{3}$, $\frac{1}{1037289919873024} a^{29} - \frac{437}{7602176} a^{24} + \frac{1372265}{7602176} a^{19} - \frac{2207085}{7602176} a^{14} + \frac{1249281}{7602176} a^{9} - \frac{500484337783}{1012978437376} a^{4}$, $\frac{1}{4149159679492096} a^{30} - \frac{101}{4149159679492096} a^{25} + \frac{221289}{30408704} a^{20} + \frac{6562451}{30408704} a^{15} - \frac{10485759}{30408704} a^{10} + \frac{1676655202461}{4051913749504} a^{5} + \frac{409350572}{3956947021}$, $\frac{1}{16596638717968384} a^{31} - \frac{101}{16596638717968384} a^{26} + \frac{221289}{121634816} a^{21} + \frac{6562451}{121634816} a^{16} + \frac{50331649}{121634816} a^{11} + \frac{1676655202461}{16207654998016} a^{6} + \frac{4366297593}{15827788084} a$, $\frac{1}{66386554871873536} a^{32} - \frac{101}{66386554871873536} a^{27} + \frac{221289}{486539264} a^{22} - \frac{236707181}{486539264} a^{17} - \frac{192937983}{486539264} a^{12} - \frac{14530999795555}{64830619992064} a^{7} - \frac{27289278575}{63311152336} a^{2}$, $\frac{1}{265546219487494144} a^{33} - \frac{101}{265546219487494144} a^{28} + \frac{221289}{1946157056} a^{23} + \frac{736371347}{1946157056} a^{18} - \frac{192937983}{1946157056} a^{13} - \frac{14530999795555}{259322479968256} a^{8} + \frac{36021873761}{253244609344} a^{3}$, $\frac{1}{1062184877949976576} a^{34} - \frac{101}{1062184877949976576} a^{29} + \frac{221289}{7784628224} a^{24} - \frac{1209785709}{7784628224} a^{19} - \frac{2139095039}{7784628224} a^{14} + \frac{244791480172701}{1037289919873024} a^{9} - \frac{217222735583}{1012978437376} a^{4}$, $\frac{1}{4248739511799906304} a^{35} - \frac{101}{4248739511799906304} a^{30} + \frac{11225}{4248739511799906304} a^{25} - \frac{10187153}{1073741824} a^{20} + \frac{259179061}{1073741824} a^{15} + \frac{1287670246596765}{4149159679492096} a^{10} - \frac{698605807655}{4051913749504} a^{5} + \frac{818678795}{3956947021}$, $\frac{1}{16994958047199625216} a^{36} - \frac{101}{16994958047199625216} a^{31} + \frac{11225}{16994958047199625216} a^{26} - \frac{10187153}{4294967296} a^{21} + \frac{1332920885}{4294967296} a^{16} - \frac{2861489432895331}{16596638717968384} a^{11} + \frac{3353307941849}{16207654998016} a^{6} - \frac{7095215247}{15827788084} a$, $\frac{1}{67979832188798500864} a^{37} - \frac{101}{67979832188798500864} a^{32} + \frac{11225}{67979832188798500864} a^{27} - \frac{10187153}{17179869184} a^{22} + \frac{1332920885}{17179869184} a^{17} - \frac{2861489432895331}{66386554871873536} a^{12} - \frac{12854347056167}{64830619992064} a^{7} - \frac{7095215247}{63311152336} a^{2}$, $\frac{1}{271919328755194003456} a^{38} - \frac{101}{271919328755194003456} a^{33} + \frac{11225}{271919328755194003456} a^{28} - \frac{10187153}{68719476736} a^{23} - \frac{15846948299}{68719476736} a^{18} - \frac{69248044304768867}{265546219487494144} a^{13} + \frac{51976272935897}{259322479968256} a^{8} - \frac{70406367583}{253244609344} a^{3}$, $\frac{1}{1087677315020776013824} a^{39} - \frac{101}{1087677315020776013824} a^{34} + \frac{11225}{1087677315020776013824} a^{29} - \frac{10187153}{274877906944} a^{24} + \frac{121592005173}{274877906944} a^{19} + \frac{196298175182725277}{1062184877949976576} a^{14} + \frac{311298752904153}{1037289919873024} a^{9} - \frac{70406367583}{1012978437376} a^{4}$

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:  \( -\frac{1}{63311152336} a^{27} - \frac{25963647845}{63311152336} a^{2} \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\zeta_{5})\), 4.0.36125.1, 5.5.390625.1, 8.0.1305015625.1, 10.10.1083264923095703125.1, \(\Q(\zeta_{25})^+\), 10.10.216652984619140625.1, 20.20.1173462893609539605677127838134765625.1, \(\Q(\zeta_{25})\), 20.0.5867314468047698028385639190673828125.1

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 $20^{2}$ $20^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
17Data not computed