sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309)
gp: K = bnfinit(y^29 - y^28 - 448*y^27 + 107*y^26 + 76138*y^25 - 43560*y^24 - 6896718*y^23 + 8492788*y^22 + 374467531*y^21 - 737137472*y^20 - 12707566814*y^19 + 34389111341*y^18 + 270515586354*y^17 - 939536322456*y^16 - 3528471156331*y^15 + 15628114907220*y^14 + 26518327124887*y^13 - 161458375653531*y^12 - 93674380846644*y^11 + 1045971567643780*y^10 - 54383106047402*y^9 - 4238937042324143*y^8 + 1725076964802557*y^7 + 10520005444822970*y^6 - 6101874655993302*y^5 - 15103434975606529*y^4 + 9152562248392343*y^3 + 10944332007104174*y^2 - 5127494290967802*y - 2711408412652309, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309)
\( x^{29} - x^{28} - 448 x^{27} + 107 x^{26} + 76138 x^{25} - 43560 x^{24} - 6896718 x^{23} + \cdots - 27\!\cdots\!09 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : $29$
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : $[29, 0]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
\(127\!\cdots\!161\)
\(\medspace = 929^{28}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(733.96\)
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: OK = ring_of_integers(K);
(1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant : $929^{28/29}\approx 733.956539804396$
Ramified primes :
\(929\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant(OK))
Discriminant root field : \(\Q\)
$\Aut(K/\Q)$
$=$
$\Gal(K/\Q)$ :
$C_{29}$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over $\Q$. Conductor : \(929\) Dirichlet character group :
$\lbrace$$\chi_{929}(1,·)$ , $\chi_{929}(261,·)$ , $\chi_{929}(454,·)$ , $\chi_{929}(72,·)$ , $\chi_{929}(521,·)$ , $\chi_{929}(524,·)$ , $\chi_{929}(719,·)$ , $\chi_{929}(400,·)$ , $\chi_{929}(148,·)$ , $\chi_{929}(537,·)$ , $\chi_{929}(539,·)$ , $\chi_{929}(20,·)$ , $\chi_{929}(352,·)$ , $\chi_{929}(673,·)$ , $\chi_{929}(347,·)$ , $\chi_{929}(511,·)$ , $\chi_{929}(807,·)$ , $\chi_{929}(173,·)$ , $\chi_{929}(304,·)$ , $\chi_{929}(561,·)$ , $\chi_{929}(437,·)$ , $\chi_{929}(201,·)$ , $\chi_{929}(568,·)$ , $\chi_{929}(212,·)$ , $\chi_{929}(506,·)$ , $\chi_{929}(379,·)$ , $\chi_{929}(445,·)$ , $\chi_{929}(830,·)$ , $\chi_{929}(575,·)$ $\rbrace$ This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{101}a^{24}+\frac{13}{101}a^{23}-\frac{1}{101}a^{22}-\frac{4}{101}a^{21}+\frac{13}{101}a^{20}+\frac{4}{101}a^{19}-\frac{32}{101}a^{18}-\frac{49}{101}a^{17}+\frac{30}{101}a^{16}-\frac{9}{101}a^{15}-\frac{32}{101}a^{14}-\frac{20}{101}a^{13}-\frac{22}{101}a^{12}-\frac{26}{101}a^{11}+\frac{17}{101}a^{10}-\frac{11}{101}a^{9}-\frac{13}{101}a^{8}-\frac{34}{101}a^{7}-\frac{23}{101}a^{6}-\frac{40}{101}a^{5}-\frac{10}{101}a^{4}-\frac{30}{101}a^{3}-\frac{28}{101}a^{2}-\frac{41}{101}a-\frac{39}{101}$, $\frac{1}{101}a^{25}+\frac{32}{101}a^{23}+\frac{9}{101}a^{22}-\frac{36}{101}a^{21}+\frac{37}{101}a^{20}+\frac{17}{101}a^{19}-\frac{37}{101}a^{18}-\frac{40}{101}a^{17}+\frac{5}{101}a^{16}-\frac{16}{101}a^{15}-\frac{8}{101}a^{14}+\frac{36}{101}a^{13}-\frac{43}{101}a^{12}-\frac{49}{101}a^{11}-\frac{30}{101}a^{10}+\frac{29}{101}a^{9}+\frac{34}{101}a^{8}+\frac{15}{101}a^{7}-\frac{44}{101}a^{6}+\frac{5}{101}a^{5}-\frac{1}{101}a^{4}-\frac{42}{101}a^{3}+\frac{20}{101}a^{2}-\frac{11}{101}a+\frac{2}{101}$, $\frac{1}{101}a^{26}-\frac{3}{101}a^{23}-\frac{4}{101}a^{22}-\frac{37}{101}a^{21}+\frac{5}{101}a^{20}+\frac{37}{101}a^{19}-\frac{26}{101}a^{18}-\frac{43}{101}a^{17}+\frac{34}{101}a^{16}-\frac{23}{101}a^{15}+\frac{50}{101}a^{14}-\frac{9}{101}a^{13}+\frac{49}{101}a^{12}-\frac{6}{101}a^{11}-\frac{10}{101}a^{10}-\frac{18}{101}a^{9}+\frac{27}{101}a^{8}+\frac{34}{101}a^{7}+\frac{34}{101}a^{6}-\frac{34}{101}a^{5}-\frac{25}{101}a^{4}-\frac{30}{101}a^{3}-\frac{24}{101}a^{2}+\frac{1}{101}a+\frac{36}{101}$, $\frac{1}{19897}a^{27}+\frac{63}{19897}a^{26}-\frac{77}{19897}a^{25}-\frac{49}{19897}a^{24}+\frac{5027}{19897}a^{23}+\frac{6033}{19897}a^{22}-\frac{7349}{19897}a^{21}+\frac{4985}{19897}a^{20}-\frac{8682}{19897}a^{19}-\frac{5743}{19897}a^{18}-\frac{8249}{19897}a^{17}-\frac{1262}{19897}a^{16}-\frac{460}{19897}a^{15}+\frac{3411}{19897}a^{14}-\frac{9743}{19897}a^{13}-\frac{8958}{19897}a^{12}+\frac{7207}{19897}a^{11}-\frac{4776}{19897}a^{10}-\frac{3642}{19897}a^{9}-\frac{2507}{19897}a^{8}+\frac{9150}{19897}a^{7}+\frac{9079}{19897}a^{6}+\frac{8681}{19897}a^{5}-\frac{9855}{19897}a^{4}-\frac{6289}{19897}a^{3}-\frac{46}{19897}a^{2}-\frac{299}{19897}a+\frac{6130}{19897}$, $\frac{1}{10\cdots 27}a^{28}-\frac{82\cdots 22}{10\cdots 27}a^{27}-\frac{37\cdots 99}{10\cdots 27}a^{26}+\frac{39\cdots 22}{10\cdots 27}a^{25}-\frac{50\cdots 45}{10\cdots 27}a^{24}+\frac{32\cdots 26}{10\cdots 27}a^{23}-\frac{47\cdots 69}{10\cdots 27}a^{22}+\frac{40\cdots 13}{10\cdots 27}a^{21}+\frac{26\cdots 88}{10\cdots 27}a^{20}+\frac{96\cdots 10}{10\cdots 27}a^{19}-\frac{16\cdots 31}{10\cdots 27}a^{18}+\frac{40\cdots 00}{10\cdots 27}a^{17}+\frac{14\cdots 22}{10\cdots 27}a^{16}+\frac{47\cdots 59}{10\cdots 27}a^{15}+\frac{22\cdots 91}{10\cdots 27}a^{14}-\frac{21\cdots 85}{10\cdots 27}a^{13}+\frac{93\cdots 97}{10\cdots 27}a^{12}+\frac{37\cdots 61}{10\cdots 27}a^{11}+\frac{13\cdots 18}{10\cdots 27}a^{10}+\frac{12\cdots 48}{10\cdots 27}a^{9}+\frac{26\cdots 62}{10\cdots 27}a^{8}-\frac{43\cdots 97}{10\cdots 27}a^{7}-\frac{32\cdots 00}{10\cdots 27}a^{6}-\frac{16\cdots 66}{10\cdots 27}a^{5}+\frac{39\cdots 08}{10\cdots 27}a^{4}-\frac{22\cdots 33}{10\cdots 27}a^{3}+\frac{39\cdots 48}{10\cdots 27}a^{2}-\frac{24\cdots 93}{10\cdots 27}a-\frac{52\cdots 73}{10\cdots 27}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : not computed
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : not computed
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : $28$
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
\( -1 \)
(order $2$)
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 : not computed
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 : not computed
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{29}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{127186199976511404062972685561977327455002805301971051425559672113022697399314124161}}\cr\mathstrut & \text{
some values not computed }
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309)
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))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309, 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)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309);
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)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = polynomial_ring(QQ); K, a = number_field(x^29 - x^28 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309);
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))))
$C_{29}$ (as 29T1 ):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K);
degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
The extension is primitive: there are no intermediate fields
between this field and $\Q$.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(L)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
$29$
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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]
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
