Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29)
gp: K = bnfinit(y^28 - 29*y^26 + 377*y^24 - 2900*y^22 + 14674*y^20 - 51359*y^18 + 127281*y^16 - 224808*y^14 + 281010*y^12 - 243542*y^10 + 140998*y^8 - 51272*y^6 + 10556*y^4 - 1015*y^2 + 29, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29)
\( x^{28} - 29 x^{26} + 377 x^{24} - 2900 x^{22} + 14674 x^{20} - 51359 x^{18} + 127281 x^{16} - 224808 x^{14} + \cdots + 29 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[28, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(819569564076950716311987772907236898045117333504\)
\(\medspace = 2^{28}\cdot 29^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.43\) | 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: | $2\cdot 29^{27/28}\approx 51.427983232816544$ | ||
| Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(116=2^{2}\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(3,·)$, $\chi_{116}(5,·)$, $\chi_{116}(65,·)$, $\chi_{116}(9,·)$, $\chi_{116}(79,·)$, $\chi_{116}(11,·)$, $\chi_{116}(109,·)$, $\chi_{116}(13,·)$, $\chi_{116}(15,·)$, $\chi_{116}(81,·)$, $\chi_{116}(75,·)$, $\chi_{116}(19,·)$, $\chi_{116}(25,·)$, $\chi_{116}(27,·)$, $\chi_{116}(93,·)$, $\chi_{116}(31,·)$, $\chi_{116}(33,·)$, $\chi_{116}(99,·)$, $\chi_{116}(39,·)$, $\chi_{116}(43,·)$, $\chi_{116}(45,·)$, $\chi_{116}(47,·)$, $\chi_{116}(49,·)$, $\chi_{116}(53,·)$, $\chi_{116}(55,·)$, $\chi_{116}(57,·)$, $\chi_{116}(95,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
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
Trivial group, which has order $1$ (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: | $27$ | 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: |
$a^{6}-6a^{4}+9a^{2}-2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{26}-27a^{24}+324a^{22}-2276a^{20}+10374a^{18}-32130a^{16}+68817a^{14}-101727a^{12}+101763a^{10}-66197a^{8}+26181a^{6}-5643a^{4}+545a^{2}-15$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24753a^{12}-27468a^{10}+19359a^{8}-8120a^{6}+1821a^{4}-180a^{2}+5$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8009a^{12}+11023a^{10}-9492a^{8}+4831a^{6}-1315a^{4}+158a^{2}-6$, $a^{22}-22a^{20}+210a^{18}-1140a^{16}+3876a^{14}-8568a^{12}+12375a^{10}-11430a^{8}+6399a^{6}-1947a^{4}+255a^{2}-9$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{26}-27a^{24}+324a^{22}-2277a^{20}+10395a^{18}-32319a^{16}+69768a^{14}-104652a^{12}+107406a^{10}-72930a^{8}+30888a^{6}-7371a^{4}+819a^{2}-27$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{24}-24a^{22}+253a^{20}-1540a^{18}+5984a^{16}-15489a^{14}+27041a^{12}-31538a^{10}+23815a^{8}-10977a^{6}+2787a^{4}-318a^{2}+9$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-a^{19}-1520a^{18}+19a^{17}+5814a^{16}-152a^{15}-14688a^{14}+665a^{13}+24752a^{12}-1729a^{11}-27456a^{10}+2717a^{9}+19305a^{8}-2508a^{7}-8008a^{6}+1254a^{5}+1716a^{4}-285a^{3}-144a^{2}+19a+2$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a+1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}-a+2$, $a^{14}-14a^{12}+77a^{10}-a^{9}-210a^{8}+9a^{7}+294a^{6}-27a^{5}-196a^{4}+30a^{3}+49a^{2}-9a-2$, $a^{23}-23a^{21}+230a^{19}-a^{18}-1311a^{17}+18a^{16}+4692a^{15}-135a^{14}-10948a^{13}+546a^{12}+16744a^{11}-1287a^{10}-16445a^{9}+1782a^{8}+9867a^{7}-1386a^{6}-3289a^{5}+540a^{4}+506a^{3}-81a^{2}-23a+2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a-1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}-a^{16}+69768a^{15}+16a^{14}-104652a^{13}-104a^{12}+107406a^{11}+352a^{10}-72930a^{9}-660a^{8}+30888a^{7}+672a^{6}-7371a^{5}-336a^{4}+819a^{3}+64a^{2}-27a-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{12}-12a^{10}+54a^{8}-a^{7}-112a^{6}+7a^{5}+105a^{4}-14a^{3}-36a^{2}+7a+2$, $a^{8}+a^{7}-8a^{6}-7a^{5}+20a^{4}+14a^{3}-16a^{2}-7a+2$, $a^{4}-4a^{2}-a+2$, $a^{18}-18a^{16}+135a^{14}-a^{13}-546a^{12}+13a^{11}+1287a^{10}-65a^{9}-1782a^{8}+156a^{7}+1386a^{6}-182a^{5}-540a^{4}+91a^{3}+81a^{2}-13a-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-a^{18}-32319a^{17}+18a^{16}+69768a^{15}-135a^{14}-104652a^{13}+546a^{12}+107406a^{11}-1287a^{10}-72929a^{9}+1782a^{8}+30879a^{7}-1386a^{6}-7344a^{5}+540a^{4}+789a^{3}-81a^{2}-18a+1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a+1$
| 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: | \( 822416597953191.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 822416597953191.9 \cdot 1}{2\cdot\sqrt{819569564076950716311987772907236898045117333504}}\cr\approx \mathstrut & 0.121929512845963 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29)
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^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29, 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^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29);
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^28 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29);
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
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 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.390224.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\) |
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 | $28$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | $28$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{7}$ | $28$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | $28$ | $28$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | $28$ | $28$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $28$ | $2$ | $14$ | $28$ | |||
|
\(29\)
| Deg $28$ | $28$ | $1$ | $27$ |