Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 5)
gp: K = bnfinit(y^25 - 5*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x - 5)
\( x^{25} - 5x - 5 \)
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: |
\(4896471684322422898509807949247762262821197509765625\)
\(\medspace = 5^{25}\cdot 269\cdot 61\!\cdots\!21\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(116.84\) | 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: | not computed | ||
| Ramified primes: |
\(5\), \(269\), \(61077\!\cdots\!67321\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{82149\!\cdots\!46745}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a+1$, $a^{24}+a^{22}+a^{20}+a^{19}+2a^{18}+2a^{17}+2a^{16}+3a^{15}+a^{14}+a^{13}-a^{12}-a^{10}-a^{9}-3a^{8}-2a^{7}-6a^{6}-7a^{5}-10a^{4}-7a^{3}-6a^{2}-2a-6$, $2a^{24}+2a^{23}+2a^{21}-a^{19}-2a^{17}+a^{15}+5a^{13}+4a^{12}+3a^{11}+7a^{10}+2a^{9}+a^{7}-4a^{6}-2a^{5}-2a^{4}-2a^{3}+8a^{2}+7a+1$, $7a^{24}+7a^{23}-14a^{22}+23a^{21}-21a^{20}+12a^{19}-13a^{17}+27a^{16}-26a^{15}+25a^{14}-3a^{13}-3a^{12}+32a^{11}-32a^{10}+34a^{9}-18a^{8}+a^{7}+25a^{6}-41a^{5}+46a^{4}-26a^{3}+26a^{2}+33a-64$, $3a^{24}-13a^{23}+6a^{22}-16a^{21}+9a^{20}-18a^{19}+11a^{18}-20a^{17}+12a^{16}-21a^{15}+14a^{14}-18a^{13}+20a^{12}-9a^{11}+31a^{10}+4a^{9}+42a^{8}+13a^{7}+43a^{6}+9a^{5}+28a^{4}-10a^{3}+a^{2}-35a-39$, $43a^{24}-37a^{23}+17a^{22}+10a^{21}-37a^{20}+54a^{19}-53a^{18}+35a^{17}-a^{16}-32a^{15}+60a^{14}-66a^{13}+56a^{12}-23a^{11}-16a^{10}+58a^{9}-84a^{8}+84a^{7}-57a^{6}+2a^{5}+52a^{4}-101a^{3}+107a^{2}-92a-184$, $8a^{24}+18a^{23}+6a^{22}+9a^{21}-2a^{20}-5a^{19}-15a^{18}-23a^{17}-29a^{16}-39a^{15}-40a^{14}-48a^{13}-42a^{12}-46a^{11}-29a^{10}-26a^{9}+11a^{7}+44a^{6}+57a^{5}+94a^{4}+106a^{3}+136a^{2}+138a+111$, $27a^{24}+12a^{23}-35a^{22}+18a^{21}+16a^{20}-28a^{19}+11a^{18}-17a^{16}+45a^{15}-20a^{14}-55a^{13}+86a^{12}+4a^{11}-126a^{10}+82a^{9}+81a^{8}-144a^{7}+40a^{6}+122a^{5}-141a^{4}-2a^{3}+134a^{2}-147a-156$, $10a^{24}-8a^{23}-8a^{22}+15a^{21}-9a^{20}+12a^{19}-11a^{18}-9a^{17}+19a^{16}-13a^{15}+21a^{14}-22a^{13}-7a^{12}+20a^{11}-10a^{10}+30a^{9}-44a^{8}+2a^{7}+21a^{6}+3a^{5}+24a^{4}-66a^{3}+18a^{2}+26a-39$, $101a^{24}-72a^{23}+91a^{22}-129a^{21}+111a^{20}-145a^{19}+148a^{18}-101a^{17}+183a^{16}-193a^{15}+77a^{14}-165a^{13}+228a^{12}-92a^{11}+135a^{10}-191a^{9}+103a^{8}-175a^{7}+131a^{6}-15a^{5}+241a^{4}-166a^{3}-123a^{2}-198a-254$, $473a^{24}-362a^{23}-371a^{22}+594a^{21}+151a^{20}-770a^{19}+199a^{18}+798a^{17}-618a^{16}-628a^{15}+1029a^{14}+222a^{13}-1292a^{12}+359a^{11}+1325a^{10}-1055a^{9}-1031a^{8}+1723a^{7}+377a^{6}-2195a^{5}+634a^{4}+2237a^{3}-1829a^{2}-1706a+579$, $7a^{24}+6a^{23}-7a^{22}-12a^{21}+8a^{20}+9a^{19}-10a^{18}-13a^{17}+6a^{16}+17a^{15}-11a^{14}-20a^{13}+13a^{12}+25a^{11}-5a^{10}-31a^{9}+12a^{8}+40a^{7}-8a^{6}-37a^{5}+8a^{4}+53a^{3}-9a^{2}-64a-26$
| 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: | \( 7307953282555373.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 7307953282555373.0 \cdot 1}{2\cdot\sqrt{4896471684322422898509807949247762262821197509765625}}\cr\approx \mathstrut & 0.395378053501284 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 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*x - 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*x - 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*x - 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
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 non-solvable group of order 15511210043330985984000000 |
| The 1958 conjugacy class representatives for $S_{25}$ |
| Character table for $S_{25}$ |
Intermediate fields
| 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(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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.7.0.1}{7} }^{2}$ | $16{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 $25$ | $25$ | $1$ | $25$ | |||
|
\(269\)
| $\Q_{269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(610\!\cdots\!321\)
| $\Q_{61\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |