sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784)
gp: K = bnfinit(y^28 - 174*y^26 + 13572*y^24 - 626400*y^22 + 19017504*y^20 - 399367584*y^18 + 5938422336*y^16 - 62931852288*y^14 + 471988892160*y^12 - 2454342239232*y^10 + 8525609883648*y^8 - 18601330655232*y^6 + 22978114338816*y^4 - 13256604426240*y^2 + 2272560758784, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784)
\( x^{28} - 174 x^{26} + 13572 x^{24} - 626400 x^{22} + 19017504 x^{20} - 399367584 x^{18} + \cdots + 2272560758784 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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 :
\(64224883807413352704377353768048295264353596334281414672384\)
\(\medspace = 2^{42}\cdot 3^{14}\cdot 29^{27}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(125.97\)
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 : $2^{3/2}3^{1/2}29^{27/28}\approx 125.9723174208094$
Ramified primes :
\(2\), \(3\), \(29\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant(OK))
Discriminant root field : \(\Q(\sqrt{29}) \)
$\Aut(K/\Q)$
$=$
$\Gal(K/\Q)$ :
$C_{28}$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over $\Q$. Conductor : \(696=2^{3}\cdot 3\cdot 29\) Dirichlet character group :
$\lbrace$$\chi_{696}(1,·)$ , $\chi_{696}(361,·)$ , $\chi_{696}(389,·)$ , $\chi_{696}(673,·)$ , $\chi_{696}(265,·)$ , $\chi_{696}(241,·)$ , $\chi_{696}(269,·)$ , $\chi_{696}(77,·)$ , $\chi_{696}(461,·)$ , $\chi_{696}(529,·)$ , $\chi_{696}(533,·)$ , $\chi_{696}(121,·)$ , $\chi_{696}(25,·)$ , $\chi_{696}(101,·)$ , $\chi_{696}(221,·)$ , $\chi_{696}(485,·)$ , $\chi_{696}(653,·)$ , $\chi_{696}(289,·)$ , $\chi_{696}(677,·)$ , $\chi_{696}(49,·)$ , $\chi_{696}(169,·)$ , $\chi_{696}(365,·)$ , $\chi_{696}(625,·)$ , $\chi_{696}(437,·)$ , $\chi_{696}(457,·)$ , $\chi_{696}(313,·)$ , $\chi_{696}(293,·)$ , $\chi_{696}(317,·)$ $\rbrace$ This is not a CM field . This field has no CM subfields.
$1$, $a$, $\frac{1}{6}a^{2}$, $\frac{1}{6}a^{3}$, $\frac{1}{36}a^{4}$, $\frac{1}{36}a^{5}$, $\frac{1}{216}a^{6}$, $\frac{1}{216}a^{7}$, $\frac{1}{1296}a^{8}$, $\frac{1}{1296}a^{9}$, $\frac{1}{7776}a^{10}$, $\frac{1}{7776}a^{11}$, $\frac{1}{46656}a^{12}$, $\frac{1}{46656}a^{13}$, $\frac{1}{279936}a^{14}$, $\frac{1}{279936}a^{15}$, $\frac{1}{1679616}a^{16}$, $\frac{1}{1679616}a^{17}$, $\frac{1}{10077696}a^{18}$, $\frac{1}{10077696}a^{19}$, $\frac{1}{60466176}a^{20}$, $\frac{1}{60466176}a^{21}$, $\frac{1}{362797056}a^{22}$, $\frac{1}{362797056}a^{23}$, $\frac{1}{2176782336}a^{24}$, $\frac{1}{2176782336}a^{25}$, $\frac{1}{13060694016}a^{26}$, $\frac{1}{13060694016}a^{27}$
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 : $27$
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 : 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)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{64224883807413352704377353768048295264353596334281414672384}}\cr\mathstrut & \text{
some values not computed }
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784)
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^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784, 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^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784);
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^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784);
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))))
$C_{28}$ (as 28T1 ):
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)
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(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
R
R
${\href{/padicField/5.14.0.1}{14} }^{2}$
${\href{/padicField/7.7.0.1}{7} }^{4}$
$28$
${\href{/padicField/13.7.0.1}{7} }^{4}$
${\href{/padicField/17.4.0.1}{4} }^{7}$
$28$
${\href{/padicField/23.14.0.1}{14} }^{2}$
R
$28$
$28$
${\href{/padicField/41.4.0.1}{4} }^{7}$
$28$
$28$
${\href{/padicField/53.7.0.1}{7} }^{4}$
${\href{/padicField/59.1.0.1}{1} }^{28}$
In the table, R denotes a ramified prime.
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)
