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Properties

Label 30.30.8432565464...6553.1
Degree $30$
Signature $[30, 0]$
Discriminant $7^{15}\cdot 31^{29}$
Root discriminant $73.15$
Ramified primes $7, 31$
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![90707, -1106515, 1106515, 19209645, -19209645, -101671507, 101671507, 235068845, -235068845, -298103379, 298103379, 235068845, -235068845, -123797075, 123797075, 45382573, -45382573, -11839955, 11839955, 2214701, -2214701, -295059, 295059, 27341, -27341, -1675, 1675, 61, -61, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 61*x^28 + 61*x^27 + 1675*x^26 - 1675*x^25 - 27341*x^24 + 27341*x^23 + 295059*x^22 - 295059*x^21 - 2214701*x^20 + 2214701*x^19 + 11839955*x^18 - 11839955*x^17 - 45382573*x^16 + 45382573*x^15 + 123797075*x^14 - 123797075*x^13 - 235068845*x^12 + 235068845*x^11 + 298103379*x^10 - 298103379*x^9 - 235068845*x^8 + 235068845*x^7 + 101671507*x^6 - 101671507*x^5 - 19209645*x^4 + 19209645*x^3 + 1106515*x^2 - 1106515*x + 90707)
 
gp: K = bnfinit(x^30 - x^29 - 61*x^28 + 61*x^27 + 1675*x^26 - 1675*x^25 - 27341*x^24 + 27341*x^23 + 295059*x^22 - 295059*x^21 - 2214701*x^20 + 2214701*x^19 + 11839955*x^18 - 11839955*x^17 - 45382573*x^16 + 45382573*x^15 + 123797075*x^14 - 123797075*x^13 - 235068845*x^12 + 235068845*x^11 + 298103379*x^10 - 298103379*x^9 - 235068845*x^8 + 235068845*x^7 + 101671507*x^6 - 101671507*x^5 - 19209645*x^4 + 19209645*x^3 + 1106515*x^2 - 1106515*x + 90707, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} - 61 x^{28} + 61 x^{27} + 1675 x^{26} - 1675 x^{25} - 27341 x^{24} + 27341 x^{23} + 295059 x^{22} - 295059 x^{21} - 2214701 x^{20} + 2214701 x^{19} + 11839955 x^{18} - 11839955 x^{17} - 45382573 x^{16} + 45382573 x^{15} + 123797075 x^{14} - 123797075 x^{13} - 235068845 x^{12} + 235068845 x^{11} + 298103379 x^{10} - 298103379 x^{9} - 235068845 x^{8} + 235068845 x^{7} + 101671507 x^{6} - 101671507 x^{5} - 19209645 x^{4} + 19209645 x^{3} + 1106515 x^{2} - 1106515 x + 90707 \)

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:  \(84325654640516545862951735142009373142681207160970656553=7^{15}\cdot 31^{29}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.15$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(217=7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{217}(64,·)$, $\chi_{217}(1,·)$, $\chi_{217}(134,·)$, $\chi_{217}(71,·)$, $\chi_{217}(8,·)$, $\chi_{217}(162,·)$, $\chi_{217}(139,·)$, $\chi_{217}(204,·)$, $\chi_{217}(13,·)$, $\chi_{217}(78,·)$, $\chi_{217}(209,·)$, $\chi_{217}(146,·)$, $\chi_{217}(83,·)$, $\chi_{217}(216,·)$, $\chi_{217}(153,·)$, $\chi_{217}(27,·)$, $\chi_{217}(34,·)$, $\chi_{217}(36,·)$, $\chi_{217}(6,·)$, $\chi_{217}(167,·)$, $\chi_{217}(104,·)$, $\chi_{217}(169,·)$, $\chi_{217}(48,·)$, $\chi_{217}(113,·)$, $\chi_{217}(50,·)$, $\chi_{217}(211,·)$, $\chi_{217}(181,·)$, $\chi_{217}(55,·)$, $\chi_{217}(183,·)$, $\chi_{217}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7193} a^{16} - \frac{1777}{7193} a^{15} - \frac{32}{7193} a^{14} + \frac{2959}{7193} a^{13} + \frac{416}{7193} a^{12} + \frac{457}{7193} a^{11} - \frac{2816}{7193} a^{10} - \frac{3592}{7193} a^{9} + \frac{3367}{7193} a^{8} + \frac{1947}{7193} a^{7} + \frac{75}{7193} a^{6} + \frac{1908}{7193} a^{5} - \frac{75}{7193} a^{4} + \frac{3382}{7193} a^{3} - \frac{999}{7193} a^{2} + \frac{2358}{7193} a + \frac{512}{7193}$, $\frac{1}{7193} a^{17} - \frac{34}{7193} a^{15} - \frac{3554}{7193} a^{14} + \frac{476}{7193} a^{13} - \frac{1190}{7193} a^{12} - \frac{3536}{7193} a^{11} - \frac{1296}{7193} a^{10} + \frac{574}{7193} a^{9} + \frac{530}{7193} a^{8} + \frac{61}{7193} a^{7} - \frac{1484}{7193} a^{6} + \frac{2538}{7193} a^{5} - \frac{419}{7193} a^{4} + \frac{2660}{7193} a^{3} - \frac{3387}{7193} a^{2} - \frac{2841}{7193} a + \frac{3506}{7193}$, $\frac{1}{7193} a^{18} + \frac{765}{7193} a^{15} - \frac{612}{7193} a^{14} - \frac{1286}{7193} a^{13} + \frac{3415}{7193} a^{12} - \frac{144}{7193} a^{11} - \frac{1661}{7193} a^{10} + \frac{683}{7193} a^{9} - \frac{549}{7193} a^{8} - \frac{23}{7193} a^{7} - \frac{2105}{7193} a^{6} - \frac{284}{7193} a^{5} + \frac{110}{7193} a^{4} - \frac{3487}{7193} a^{3} - \frac{842}{7193} a^{2} - \frac{2638}{7193} a + \frac{3022}{7193}$, $\frac{1}{7193} a^{19} - \frac{684}{7193} a^{15} + \frac{1615}{7193} a^{14} - \frac{1618}{7193} a^{13} - \frac{1892}{7193} a^{12} + \frac{1191}{7193} a^{11} - \frac{2977}{7193} a^{10} - \frac{395}{7193} a^{9} - \frac{684}{7193} a^{8} - \frac{2609}{7193} a^{7} - \frac{115}{7193} a^{6} + \frac{669}{7193} a^{5} + \frac{3537}{7193} a^{4} + \frac{1408}{7193} a^{3} - \frac{861}{7193} a^{2} - \frac{2598}{7193} a - \frac{3258}{7193}$, $\frac{1}{7193} a^{20} + \frac{1764}{7193} a^{15} - \frac{1927}{7193} a^{14} + \frac{831}{7193} a^{13} - \frac{1985}{7193} a^{12} + \frac{312}{7193} a^{11} + \frac{1185}{7193} a^{10} + \frac{2394}{7193} a^{9} - \frac{1341}{7193} a^{8} + \frac{928}{7193} a^{7} + \frac{1618}{7193} a^{6} - \frac{517}{7193} a^{5} + \frac{459}{7193} a^{4} + \frac{3474}{7193} a^{3} - \frac{2579}{7193} a^{2} - \frac{1618}{7193} a - \frac{2249}{7193}$, $\frac{1}{7193} a^{21} - \frac{3447}{7193} a^{15} - \frac{265}{7193} a^{14} + \frac{457}{7193} a^{13} + \frac{174}{7193} a^{12} + \frac{653}{7193} a^{11} - \frac{545}{7193} a^{10} - \frac{2086}{7193} a^{9} + \frac{2958}{7193} a^{8} - \frac{1829}{7193} a^{7} - \frac{3343}{7193} a^{6} + \frac{1071}{7193} a^{5} - \frac{893}{7193} a^{4} + \frac{1763}{7193} a^{3} - \frac{1667}{7193} a^{2} + \frac{2986}{7193} a + \frac{3150}{7193}$, $\frac{1}{7193} a^{22} + \frac{2852}{7193} a^{15} - \frac{1952}{7193} a^{14} + \frac{173}{7193} a^{13} + \frac{3198}{7193} a^{12} - \frac{533}{7193} a^{11} + \frac{1712}{7193} a^{10} + \frac{487}{7193} a^{9} + \frac{1911}{7193} a^{8} - \frac{3103}{7193} a^{7} + \frac{648}{7193} a^{6} + \frac{1581}{7193} a^{5} + \frac{2186}{7193} a^{4} + \frac{3427}{7193} a^{3} - \frac{2313}{7193} a^{2} + \frac{3086}{7193} a + \frac{2579}{7193}$, $\frac{1}{7193} a^{23} + \frac{2180}{7193} a^{15} - \frac{2072}{7193} a^{14} + \frac{1519}{7193} a^{13} - \frac{120}{7193} a^{12} + \frac{281}{7193} a^{11} - \frac{2862}{7193} a^{10} + \frac{3463}{7193} a^{9} - \frac{3132}{7193} a^{8} + \frac{800}{7193} a^{7} + \frac{3471}{7193} a^{6} - \frac{1522}{7193} a^{5} + \frac{1537}{7193} a^{4} - \frac{1964}{7193} a^{3} - \frac{3387}{7193} a^{2} + \frac{3018}{7193} a - \frac{45}{7193}$, $\frac{1}{7193} a^{24} + \frac{1954}{7193} a^{15} - \frac{651}{7193} a^{14} + \frac{1381}{7193} a^{13} - \frac{281}{7193} a^{12} + \frac{705}{7193} a^{11} - \frac{479}{7193} a^{10} + \frac{1444}{7193} a^{9} - \frac{2400}{7193} a^{8} + \frac{2881}{7193} a^{7} + \frac{417}{7193} a^{6} - \frac{349}{7193} a^{5} + \frac{3290}{7193} a^{4} - \frac{3322}{7193} a^{3} + \frac{1359}{7193} a^{2} + \frac{2510}{7193} a - \frac{1245}{7193}$, $\frac{1}{7193} a^{25} - \frac{2612}{7193} a^{15} - \frac{828}{7193} a^{14} + \frac{1005}{7193} a^{13} + \frac{650}{7193} a^{12} - \frac{1525}{7193} a^{11} + \frac{1263}{7193} a^{10} + \frac{3193}{7193} a^{9} - \frac{1835}{7193} a^{8} + \frac{1076}{7193} a^{7} - \frac{3039}{7193} a^{6} + \frac{1032}{7193} a^{5} - \frac{632}{7193} a^{4} + \frac{3298}{7193} a^{3} - \frac{1940}{7193} a^{2} + \frac{1936}{7193} a - \frac{621}{7193}$, $\frac{1}{7193} a^{26} - \frac{2867}{7193} a^{15} - \frac{3456}{7193} a^{14} - \frac{2917}{7193} a^{13} - \frac{1076}{7193} a^{12} + \frac{909}{7193} a^{11} - \frac{953}{7193} a^{10} + \frac{2726}{7193} a^{9} - \frac{1359}{7193} a^{8} - \frac{2926}{7193} a^{7} + \frac{2721}{7193} a^{6} - \frac{1685}{7193} a^{5} + \frac{1609}{7193} a^{4} - \frac{1160}{7193} a^{3} - \frac{3586}{7193} a^{2} + \frac{1267}{7193} a - \frac{554}{7193}$, $\frac{1}{7193} a^{27} + \frac{1722}{7193} a^{15} - \frac{1152}{7193} a^{14} + \frac{1830}{7193} a^{13} - \frac{457}{7193} a^{12} + \frac{140}{7193} a^{11} - \frac{200}{7193} a^{10} + \frac{753}{7193} a^{9} - \frac{2743}{7193} a^{8} + \frac{3002}{7193} a^{7} - \frac{2450}{7193} a^{6} - \frac{2028}{7193} a^{5} - \frac{395}{7193} a^{4} - \frac{3556}{7193} a^{3} - \frac{52}{7193} a^{2} - \frac{1588}{7193} a + \frac{532}{7193}$, $\frac{1}{7193} a^{28} + \frac{1817}{7193} a^{15} - \frac{610}{7193} a^{14} - \frac{3211}{7193} a^{13} + \frac{3088}{7193} a^{12} - \frac{3117}{7193} a^{11} + \frac{1823}{7193} a^{10} - \frac{3299}{7193} a^{9} + \frac{2586}{7193} a^{8} - \frac{3246}{7193} a^{7} - \frac{1704}{7193} a^{6} + \frac{1230}{7193} a^{5} + \frac{3313}{7193} a^{4} + \frac{2474}{7193} a^{3} - \frac{437}{7193} a^{2} - \frac{3092}{7193} a + \frac{3075}{7193}$, $\frac{1}{7193} a^{29} - \frac{1458}{7193} a^{15} - \frac{2611}{7193} a^{14} - \frac{244}{7193} a^{13} + \frac{3469}{7193} a^{12} - \frac{1351}{7193} a^{11} - \frac{850}{7193} a^{10} - \frac{1994}{7193} a^{9} + \frac{158}{7193} a^{8} - \frac{447}{7193} a^{7} + \frac{1622}{7193} a^{6} + \frac{3503}{7193} a^{5} + \frac{2082}{7193} a^{4} - \frac{2709}{7193} a^{3} - \frac{545}{7193} a^{2} - \frac{1576}{7193} a - \frac{2407}{7193}$

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:  \( 3598429667507702300 \) (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

\(\Q(\sqrt{217}) \), 3.3.961.1, 5.5.923521.1, 6.6.9819798793.1, 10.10.444370729654397497.1, \(\Q(\zeta_{31})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{6}$ $15^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ R $30$ $15^{2}$ $15^{2}$ $30$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ $30$ $30$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $30$ $30$

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
7Data not computed
31Data not computed