Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125)
gp: K = bnfinit(y^28 - 145*y^26 + 9425*y^24 - 362500*y^22 + 9171250*y^20 - 160496875*y^18 + 1988765625*y^16 - 17563125000*y^14 + 109769531250*y^12 - 475667968750*y^10 + 1376933593750*y^8 - 2503515625000*y^6 + 2577148437500*y^4 - 1239013671875*y^2 + 177001953125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125)
\( x^{28} - 145 x^{26} + 9425 x^{24} - 362500 x^{22} + 9171250 x^{20} - 160496875 x^{18} + \cdots + 177001953125 \)
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: |
\(5002255640118107399365159746748272082794905600000000000000\)
\(\medspace = 2^{28}\cdot 5^{14}\cdot 29^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(115.00\) | 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: | $2\cdot 5^{1/2}29^{27/28}\approx 114.99646645429719$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\)
| 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: | \(580=2^{2}\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(259,·)$, $\chi_{580}(81,·)$, $\chi_{580}(519,·)$, $\chi_{580}(521,·)$, $\chi_{580}(141,·)$, $\chi_{580}(79,·)$, $\chi_{580}(401,·)$, $\chi_{580}(19,·)$, $\chi_{580}(121,·)$, $\chi_{580}(341,·)$, $\chi_{580}(279,·)$, $\chi_{580}(281,·)$, $\chi_{580}(539,·)$, $\chi_{580}(159,·)$, $\chi_{580}(161,·)$, $\chi_{580}(99,·)$, $\chi_{580}(39,·)$, $\chi_{580}(379,·)$, $\chi_{580}(361,·)$, $\chi_{580}(359,·)$, $\chi_{580}(559,·)$, $\chi_{580}(241,·)$, $\chi_{580}(181,·)$, $\chi_{580}(119,·)$, $\chi_{580}(441,·)$, $\chi_{580}(479,·)$, $\chi_{580}(381,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{125}a^{7}$, $\frac{1}{625}a^{8}$, $\frac{1}{625}a^{9}$, $\frac{1}{3125}a^{10}$, $\frac{1}{3125}a^{11}$, $\frac{1}{15625}a^{12}$, $\frac{1}{15625}a^{13}$, $\frac{1}{78125}a^{14}$, $\frac{1}{78125}a^{15}$, $\frac{1}{390625}a^{16}$, $\frac{1}{390625}a^{17}$, $\frac{1}{1953125}a^{18}$, $\frac{1}{1953125}a^{19}$, $\frac{1}{9765625}a^{20}$, $\frac{1}{9765625}a^{21}$, $\frac{1}{48828125}a^{22}$, $\frac{1}{48828125}a^{23}$, $\frac{1}{244140625}a^{24}$, $\frac{1}{244140625}a^{25}$, $\frac{1}{1220703125}a^{26}$, $\frac{1}{1220703125}a^{27}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
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: | 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)
|
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 $
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125)
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 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125, 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 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125);
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 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125);
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 $C_{28}$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.9755600.2, 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 | $28$ | R | ${\href{/padicField/7.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/13.7.0.1}{7} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{7}$ | $28$ | ${\href{/padicField/23.7.0.1}{7} }^{4}$ | R | $28$ | $28$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | $28$ | $28$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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 $28$ | $2$ | $14$ | $28$ | |||
|
\(5\)
| 5.14.7.2 | $x^{14} + 46875 x^{2} - 234375$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| 5.14.7.2 | $x^{14} + 46875 x^{2} - 234375$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(29\)
| Deg $28$ | $28$ | $1$ | $27$ |