Properties

Label 30.30.175...397.1
Degree $30$
Signature $[30, 0]$
Discriminant $1.758\times 10^{49}$
Root discriminant $43.80$
Ramified primes $7, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 30*x^28 + 29*x^27 + 405*x^26 - 377*x^25 - 3250*x^24 + 2901*x^23 + 17249*x^22 - 14697*x^21 - 63734*x^20 + 51590*x^19 + 168035*x^18 - 128611*x^17 - 318629*x^16 + 229651*x^15 + 432159*x^14 - 292608*x^13 - 411241*x^12 + 261924*x^11 + 264472*x^10 - 159873*x^9 - 107406*x^8 + 63143*x^7 + 24051*x^6 - 14609*x^5 - 2010*x^4 + 1562*x^3 - 72*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^30 - x^29 - 30*x^28 + 29*x^27 + 405*x^26 - 377*x^25 - 3250*x^24 + 2901*x^23 + 17249*x^22 - 14697*x^21 - 63734*x^20 + 51590*x^19 + 168035*x^18 - 128611*x^17 - 318629*x^16 + 229651*x^15 + 432159*x^14 - 292608*x^13 - 411241*x^12 + 261924*x^11 + 264472*x^10 - 159873*x^9 - 107406*x^8 + 63143*x^7 + 24051*x^6 - 14609*x^5 - 2010*x^4 + 1562*x^3 - 72*x^2 - 24*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, -72, 1562, -2010, -14609, 24051, 63143, -107406, -159873, 264472, 261924, -411241, -292608, 432159, 229651, -318629, -128611, 168035, 51590, -63734, -14697, 17249, 2901, -3250, -377, 405, 29, -30, -1, 1]);
 

\( x^{30} - x^{29} - 30 x^{28} + 29 x^{27} + 405 x^{26} - 377 x^{25} - 3250 x^{24} + 2901 x^{23} + 17249 x^{22} - 14697 x^{21} - 63734 x^{20} + 51590 x^{19} + 168035 x^{18} - 128611 x^{17} - 318629 x^{16} + 229651 x^{15} + 432159 x^{14} - 292608 x^{13} - 411241 x^{12} + 261924 x^{11} + 264472 x^{10} - 159873 x^{9} - 107406 x^{8} + 63143 x^{7} + 24051 x^{6} - 14609 x^{5} - 2010 x^{4} + 1562 x^{3} - 72 x^{2} - 24 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[30, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(17581401814197409148890873176573567284303576419397\)\(\medspace = 7^{25}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.80$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(67,·)$, $\chi_{77}(4,·)$, $\chi_{77}(6,·)$, $\chi_{77}(71,·)$, $\chi_{77}(9,·)$, $\chi_{77}(10,·)$, $\chi_{77}(76,·)$, $\chi_{77}(13,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(17,·)$, $\chi_{77}(19,·)$, $\chi_{77}(23,·)$, $\chi_{77}(24,·)$, $\chi_{77}(25,·)$, $\chi_{77}(68,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(40,·)$, $\chi_{77}(41,·)$, $\chi_{77}(52,·)$, $\chi_{77}(53,·)$, $\chi_{77}(54,·)$, $\chi_{77}(73,·)$, $\chi_{77}(58,·)$, $\chi_{77}(60,·)$, $\chi_{77}(61,·)$, $\chi_{77}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $29$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 864355592506536.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 864355592506536.0 \cdot 1}{2\sqrt{17581401814197409148890873176573567284303576419397}}\approx 0.110671442530690$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.22370117.1, 10.10.39630026842637.1, 15.15.886528337182930278529.1

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 $30$ $30$ $30$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $30$ $15^{2}$ $30$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
11Data not computed