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Properties

Label 30.0.615...784.1
Degree $30$
Signature $[0, 15]$
Discriminant $-6.152\times 10^{50}$
Root discriminant \(49.31\)
Ramified primes $2,31$
Class number $5084$ (GRH)
Class group [2, 2542] (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1)
 
gp: K = bnfinit(y^30 + 29*y^28 + 378*y^26 + 2925*y^24 + 14950*y^22 + 53130*y^20 + 134596*y^18 + 245157*y^16 + 319770*y^14 + 293930*y^12 + 184756*y^10 + 75582*y^8 + 18564*y^6 + 2380*y^4 + 120*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1)
 

\( x^{30} + 29 x^{28} + 378 x^{26} + 2925 x^{24} + 14950 x^{22} + 53130 x^{20} + 134596 x^{18} + 245157 x^{16} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-615215540441622698713738389172402189599059721846784\) \(\medspace = -\,2^{30}\cdot 31^{28}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(49.31\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 31^{14/15}\approx 49.31369861109494$
Ramified primes:   \(2\), \(31\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $30$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(124=2^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{124}(1,·)$, $\chi_{124}(67,·)$, $\chi_{124}(97,·)$, $\chi_{124}(5,·)$, $\chi_{124}(7,·)$, $\chi_{124}(9,·)$, $\chi_{124}(111,·)$, $\chi_{124}(109,·)$, $\chi_{124}(81,·)$, $\chi_{124}(19,·)$, $\chi_{124}(87,·)$, $\chi_{124}(25,·)$, $\chi_{124}(103,·)$, $\chi_{124}(69,·)$, $\chi_{124}(107,·)$, $\chi_{124}(33,·)$, $\chi_{124}(35,·)$, $\chi_{124}(113,·)$, $\chi_{124}(101,·)$, $\chi_{124}(39,·)$, $\chi_{124}(41,·)$, $\chi_{124}(95,·)$, $\chi_{124}(71,·)$, $\chi_{124}(45,·)$, $\chi_{124}(47,·)$, $\chi_{124}(49,·)$, $\chi_{124}(51,·)$, $\chi_{124}(121,·)$, $\chi_{124}(59,·)$, $\chi_{124}(63,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{16384}$

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

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

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{2542}$, which has order $5084$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( a^{29} + 28 a^{27} + 351 a^{25} + 2600 a^{23} + 12650 a^{21} + 42504 a^{19} + 100947 a^{17} + 170544 a^{15} + 203490 a^{13} + 167960 a^{11} + 92378 a^{9} + 31824 a^{7} + 6188 a^{5} + 560 a^{3} + 15 a \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{29}+28a^{27}+350a^{25}+2576a^{23}+12398a^{21}+40984a^{19}+95132a^{17}+155840a^{15}+178634a^{13}+140152a^{11}+72412a^{9}+23136a^{7}+4116a^{5}+336a^{3}+8a$, $a$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+450a^{7}+378a^{5}+140a^{3}+15a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{25}+25a^{23}+275a^{21}+1750a^{19}+7125a^{17}+19380a^{15}+35700a^{13}+44200a^{11}+35750a^{9}+17875a^{7}+5005a^{5}+650a^{3}+25a$, $a^{4}+4a^{2}+3$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$, $a^{21}+20a^{19}+170a^{17}+800a^{15}+2275a^{13}+4004a^{11}+4290a^{9}+2640a^{7}+825a^{5}+100a^{3}+3a$, $a^{29}+28a^{27}+351a^{25}+2600a^{23}+12649a^{21}+42483a^{19}+100758a^{17}+169592a^{15}+200550a^{13}+162228a^{11}+85382a^{9}+26720a^{7}+4186a^{5}+230a^{3}+4a$, $a^{28}+27a^{26}+324a^{24}+2277a^{22}+10396a^{20}+32338a^{18}+69920a^{16}+105316a^{14}+109122a^{12}+75581a^{10}+33230a^{8}+8407a^{6}+958a^{4}+2a^{2}-4$, $a^{22}+22a^{20}+209a^{18}+1122a^{16}+3740a^{14}+8008a^{12}+11011a^{10}+9438a^{8}+4719a^{6}+1210a^{4}+121a^{2}+1$, $a^{18}+18a^{16}+135a^{14}+546a^{12}+1287a^{10}+1782a^{8}+1386a^{6}+540a^{4}+81a^{2}+1$, $a^{3}+3a$, $a^{29}+29a^{27}+378a^{25}+2925a^{23}+14950a^{21}+53130a^{19}+134596a^{17}+245156a^{15}+319755a^{13}+293839a^{11}+184470a^{9}+75087a^{7}+18102a^{5}+2170a^{3}+84a$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4316173757.895952 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 5084}{4\cdot\sqrt{615215540441622698713738389172402189599059721846784}}\cr\approx \mathstrut & 0.207696341766930 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.961.1, 5.5.923521.1, 6.0.59105344.1, 10.0.873360422339584.1, \(\Q(\zeta_{31})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $30$ ${\href{/padicField/5.3.0.1}{3} }^{10}$ $30$ $30$ $15^{2}$ $15^{2}$ $30$ ${\href{/padicField/23.10.0.1}{10} }^{3}$ ${\href{/padicField/29.5.0.1}{5} }^{6}$ R ${\href{/padicField/37.3.0.1}{3} }^{10}$ $15^{2}$ $30$ ${\href{/padicField/47.10.0.1}{10} }^{3}$ $15^{2}$ $30$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.10.10.7$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$$2$$5$$10$$C_{10}$$[2]^{5}$
\(31\) Copy content Toggle raw display Deg $30$$15$$2$$28$