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Properties

Label 30.30.3543488449...8125.1
Degree $30$
Signature $[30, 0]$
Discriminant $5^{51}\cdot 7^{20}$
Root discriminant $56.45$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -10, 175, 305, -5775, -3807, 68440, 27675, -396240, -115475, 1287922, 277525, -2523400, -389415, 3095850, 319842, -2428495, -152550, 1234120, 42000, -409677, -6490, 88775, 515, -12350, -16, 1055, 0, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 50*x^28 + 1055*x^26 - 16*x^25 - 12350*x^24 + 515*x^23 + 88775*x^22 - 6490*x^21 - 409677*x^20 + 42000*x^19 + 1234120*x^18 - 152550*x^17 - 2428495*x^16 + 319842*x^15 + 3095850*x^14 - 389415*x^13 - 2523400*x^12 + 277525*x^11 + 1287922*x^10 - 115475*x^9 - 396240*x^8 + 27675*x^7 + 68440*x^6 - 3807*x^5 - 5775*x^4 + 305*x^3 + 175*x^2 - 10*x - 1)
 
gp: K = bnfinit(x^30 - 50*x^28 + 1055*x^26 - 16*x^25 - 12350*x^24 + 515*x^23 + 88775*x^22 - 6490*x^21 - 409677*x^20 + 42000*x^19 + 1234120*x^18 - 152550*x^17 - 2428495*x^16 + 319842*x^15 + 3095850*x^14 - 389415*x^13 - 2523400*x^12 + 277525*x^11 + 1287922*x^10 - 115475*x^9 - 396240*x^8 + 27675*x^7 + 68440*x^6 - 3807*x^5 - 5775*x^4 + 305*x^3 + 175*x^2 - 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{30} - 50 x^{28} + 1055 x^{26} - 16 x^{25} - 12350 x^{24} + 515 x^{23} + 88775 x^{22} - 6490 x^{21} - 409677 x^{20} + 42000 x^{19} + 1234120 x^{18} - 152550 x^{17} - 2428495 x^{16} + 319842 x^{15} + 3095850 x^{14} - 389415 x^{13} - 2523400 x^{12} + 277525 x^{11} + 1287922 x^{10} - 115475 x^{9} - 396240 x^{8} + 27675 x^{7} + 68440 x^{6} - 3807 x^{5} - 5775 x^{4} + 305 x^{3} + 175 x^{2} - 10 x - 1 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[30, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35434884492252294752034913472016341984272003173828125=5^{51}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.45$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(64,·)$, $\chi_{175}(1,·)$, $\chi_{175}(4,·)$, $\chi_{175}(134,·)$, $\chi_{175}(71,·)$, $\chi_{175}(9,·)$, $\chi_{175}(74,·)$, $\chi_{175}(11,·)$, $\chi_{175}(141,·)$, $\chi_{175}(79,·)$, $\chi_{175}(16,·)$, $\chi_{175}(81,·)$, $\chi_{175}(149,·)$, $\chi_{175}(86,·)$, $\chi_{175}(151,·)$, $\chi_{175}(156,·)$, $\chi_{175}(29,·)$, $\chi_{175}(144,·)$, $\chi_{175}(99,·)$, $\chi_{175}(36,·)$, $\chi_{175}(39,·)$, $\chi_{175}(169,·)$, $\chi_{175}(106,·)$, $\chi_{175}(44,·)$, $\chi_{175}(109,·)$, $\chi_{175}(46,·)$, $\chi_{175}(114,·)$, $\chi_{175}(51,·)$, $\chi_{175}(116,·)$, $\chi_{175}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{3127920622011438224102857257954752717037778153104322649} a^{29} - \frac{818445336242511988617914915736037301948327415068887621}{3127920622011438224102857257954752717037778153104322649} a^{28} - \frac{193645643853457988810375203757278012879986620308505344}{3127920622011438224102857257954752717037778153104322649} a^{27} + \frac{122032978172731528529349537932406618864967549254126933}{3127920622011438224102857257954752717037778153104322649} a^{26} + \frac{166041983238636619192515984473126759650421781054468991}{3127920622011438224102857257954752717037778153104322649} a^{25} + \frac{762170927229882025519073430077127531011485291706220420}{3127920622011438224102857257954752717037778153104322649} a^{24} - \frac{1225385551563962709975871422731001875990460824682854318}{3127920622011438224102857257954752717037778153104322649} a^{23} + \frac{1503502215892784975379812584481550005481823976812002690}{3127920622011438224102857257954752717037778153104322649} a^{22} + \frac{989895704295722654671073452870011808680271499509330326}{3127920622011438224102857257954752717037778153104322649} a^{21} - \frac{523148185920421465887477889013410226024126583628271193}{3127920622011438224102857257954752717037778153104322649} a^{20} - \frac{685804483143634809446583932302357269071525322874678030}{3127920622011438224102857257954752717037778153104322649} a^{19} - \frac{557359742418798993988546830935855703844244988541836733}{3127920622011438224102857257954752717037778153104322649} a^{18} - \frac{395222805415411007426126377525827287917645536303507426}{3127920622011438224102857257954752717037778153104322649} a^{17} + \frac{706285160656912907977673663814272853402218190518308759}{3127920622011438224102857257954752717037778153104322649} a^{16} - \frac{469345485947647147671272694638514463453310436631490092}{3127920622011438224102857257954752717037778153104322649} a^{15} + \frac{1293875246579970370882313272733981454197966875422679403}{3127920622011438224102857257954752717037778153104322649} a^{14} - \frac{1326492954259222594732664755074376772571069575014617136}{3127920622011438224102857257954752717037778153104322649} a^{13} + \frac{251818680068724966591359610891399913287436750095614677}{3127920622011438224102857257954752717037778153104322649} a^{12} + \frac{1273924738431125289442842560753898944372073151738183563}{3127920622011438224102857257954752717037778153104322649} a^{11} - \frac{32571223517041462832814441886130376097946977622972112}{3127920622011438224102857257954752717037778153104322649} a^{10} + \frac{580170180974519067156702893010808386799763546777026748}{3127920622011438224102857257954752717037778153104322649} a^{9} - \frac{4232029249076675130203684516099737484646387902032924}{3127920622011438224102857257954752717037778153104322649} a^{8} - \frac{175823257215512454921949520865504851342958541344999186}{3127920622011438224102857257954752717037778153104322649} a^{7} + \frac{85841311851891255540085432308725768113914962120567731}{3127920622011438224102857257954752717037778153104322649} a^{6} + \frac{1032752370200447680342747615109354838821047923207485578}{3127920622011438224102857257954752717037778153104322649} a^{5} - \frac{252395185812336229872975069909926502857457380679908024}{3127920622011438224102857257954752717037778153104322649} a^{4} - \frac{377719726594260333383331437031031218449889405424593864}{3127920622011438224102857257954752717037778153104322649} a^{3} - \frac{1045799120581013476095249171926043385394654769293690}{6966415639223693149449570730411475984493937980187801} a^{2} + \frac{50078666046625909806601242917662265063982322019858446}{3127920622011438224102857257954752717037778153104322649} a - \frac{638486369982411233556901956213716079731325215189781345}{3127920622011438224102857257954752717037778153104322649}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $29$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 46287634688780824 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 5.5.390625.1, 6.6.300125.1, \(\Q(\zeta_{25})^+\), 15.15.16836836874485015869140625.1

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 $30$ $30$ R R $15^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $30$ $15^{2}$ $30$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $15^{2}$ $30$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $30$ $30$ $15^{2}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$