Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5)
gp: K = bnfinit(y^25 - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5)
\( x^{25} - 5 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5293955920339377119177015629247762262821197509765625\)
\(\medspace = 5^{74}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(117.21\) | 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: | $5^{303/100}\approx 131.1834696007442$ | ||
| Ramified primes: |
\(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
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: | $12$ | 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: |
$a^{20}+2a^{15}+5a^{5}+1$, $a^{20}-a^{15}-1$, $a^{24}+2a^{23}-2a^{22}-a^{21}+a^{19}+2a^{18}-2a^{17}-a^{16}+a^{14}+2a^{13}+5a^{11}-5a^{9}+5a^{6}-5a^{4}+5a+1$, $a^{24}+2a^{23}-2a^{22}-a^{21}-4a^{20}+7a^{19}+2a^{18}-2a^{17}-8a^{16}+12a^{14}-6a^{12}-13a^{11}+12a^{10}+9a^{9}-a^{8}-12a^{7}-6a^{6}+15a^{5}+3a^{3}-15a^{2}+5a-1$, $16a^{24}-3a^{23}-11a^{22}+11a^{21}+3a^{20}-7a^{19}-6a^{18}+8a^{17}+3a^{16}-16a^{15}+a^{14}+14a^{13}-5a^{12}-21a^{11}+19a^{10}+17a^{9}-38a^{8}+13a^{7}+23a^{6}-20a^{5}-4a^{4}-2a^{3}+35a^{2}-20a-39$, $10a^{24}+11a^{23}-2a^{22}-24a^{21}-39a^{20}-34a^{19}-13a^{18}+6a^{17}+9a^{16}-a^{15}-9a^{14}-3a^{13}+15a^{12}+28a^{11}+15a^{10}-28a^{9}-75a^{8}-88a^{7}-55a^{6}-5a^{5}+21a^{4}+10a^{3}-15a^{2}-20a+9$, $123a^{24}+116a^{23}+76a^{22}+9a^{21}-72a^{20}-148a^{19}-198a^{18}-204a^{17}-161a^{16}-73a^{15}+43a^{14}+161a^{13}+250a^{12}+285a^{11}+247a^{10}+139a^{9}-16a^{8}-181a^{7}-315a^{6}-375a^{5}-335a^{4}-191a^{3}+35a^{2}+290a+504$, $374a^{24}-282a^{23}-462a^{22}+253a^{21}+557a^{20}-214a^{19}-666a^{18}+162a^{17}+774a^{16}-64a^{15}-899a^{14}-26a^{13}+999a^{12}+186a^{11}-1144a^{10}-341a^{9}+1214a^{8}+580a^{7}-1345a^{6}-800a^{5}+1387a^{4}+1144a^{3}-1455a^{2}-1450a+1429$, $19a^{24}-63a^{23}-8a^{22}+99a^{21}-19a^{20}-76a^{19}+30a^{18}-a^{17}-8a^{16}+95a^{15}-51a^{14}-143a^{13}+106a^{12}+103a^{11}-92a^{10}+a^{9}-7a^{8}-112a^{7}+160a^{6}+145a^{5}-267a^{4}-76a^{3}+210a^{2}-30a+14$, $24a^{24}-28a^{23}+11a^{22}-6a^{21}-16a^{20}+23a^{19}-44a^{18}+41a^{17}-55a^{16}+37a^{15}-36a^{14}+7a^{13}+6a^{12}-44a^{11}+50a^{10}-88a^{9}+71a^{8}-91a^{7}+50a^{6}-40a^{5}-13a^{4}+40a^{3}-100a^{2}+105a-161$, $8a^{24}-12a^{23}+9a^{22}-3a^{21}+2a^{20}-8a^{19}+14a^{18}-16a^{17}+17a^{16}-17a^{15}+11a^{14}+9a^{13}-38a^{12}+58a^{11}-60a^{10}+50a^{9}-33a^{8}+4a^{7}+40a^{6}-80a^{5}+92a^{4}-75a^{3}+45a^{2}-20a-6$, $28a^{24}+10a^{23}+39a^{22}-4a^{21}-44a^{20}-63a^{19}+36a^{18}+58a^{17}+24a^{16}-11a^{15}-21a^{14}-68a^{13}-47a^{12}+89a^{11}+106a^{10}-a^{9}-113a^{8}-32a^{7}-55a^{6}+50a^{5}+104a^{4}+151a^{3}-150a^{2}-155a-74$
| 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: | \( 20092696558859190 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 20092696558859190 \cdot 1}{2\cdot\sqrt{5293955920339377119177015629247762262821197509765625}}\cr\approx \mathstrut & 1.04545774762938 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 5)
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^25 - 5, 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^25 - 5);
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^25 - 5);
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
$C_{25}:C_{20}$ (as 25T40):
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 solvable group of order 500 |
| The 26 conjugacy class representatives for $C_{25}:C_{20}$ |
| Character table for $C_{25}:C_{20}$ is not computed |
Intermediate fields
| 5.1.1953125.1 |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$ | $20{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $25$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $25$ | $25$ | $1$ | $74$ |