Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x - 5)
gp: K = bnfinit(y^25 - 3*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 3*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x - 5)
\( x^{25} - 3x - 5 \)
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: |
\(5293954790280245387320429425326762678403863692388857\)
\(\medspace = 35846147\cdot 76867751\cdot 19\!\cdots\!81\)
| 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: | $35846147^{1/2}76867751^{1/2}1921292874428364109762304476450296181^{1/2}\approx 7.2759568376126625e+25$ | ||
| Ramified primes: |
\(35846147\), \(76867751\), \(19212\!\cdots\!96181\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{52939\!\cdots\!88857}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $3a^{24}+81a^{23}+88a^{22}+5a^{21}-98a^{20}-117a^{19}-17a^{18}+118a^{17}+153a^{16}+35a^{15}-139a^{14}-199a^{13}-63a^{12}+163a^{11}+258a^{10}+100a^{9}-188a^{8}-329a^{7}-153a^{6}+212a^{5}+420a^{4}+226a^{3}-237a^{2}-532a-332$, $5a^{24}+15a^{23}+6a^{22}-12a^{21}-20a^{20}-22a^{19}-16a^{18}+a^{17}+4a^{16}-5a^{15}-4a^{14}-5a^{13}-13a^{12}-15a^{11}-30a^{10}-51a^{9}-34a^{8}+5a^{7}+30a^{6}+38a^{5}-4a^{4}-92a^{3}-130a^{2}-91a-32$, $33a^{24}+48a^{23}-25a^{22}-96a^{21}-63a^{20}+35a^{19}+54a^{18}-50a^{17}-136a^{16}-63a^{15}+91a^{14}+115a^{13}-33a^{12}-134a^{11}-5a^{10}+216a^{9}+226a^{8}-7a^{7}-153a^{6}+34a^{5}+319a^{4}+267a^{3}-120a^{2}-333a-158$, $a^{24}+22a^{23}-22a^{22}-3a^{21}+20a^{20}-26a^{19}+5a^{18}+44a^{17}-27a^{16}-16a^{15}+49a^{14}-10a^{13}-10a^{12}+49a^{11}-34a^{10}-33a^{9}+78a^{8}-11a^{7}-69a^{6}+45a^{5}-15a^{4}-60a^{3}+73a^{2}-34a-143$, $42a^{24}-43a^{22}+44a^{21}+7a^{20}-58a^{19}+44a^{18}+19a^{17}-72a^{16}+44a^{15}+38a^{14}-85a^{13}+43a^{12}+63a^{11}-98a^{10}+35a^{9}+87a^{8}-118a^{7}+11a^{6}+111a^{5}-135a^{4}-12a^{3}+157a^{2}-130a-157$, $9a^{24}-21a^{23}+2a^{22}-25a^{21}+9a^{20}-19a^{19}+22a^{18}-a^{17}+31a^{16}+3a^{15}+30a^{14}-10a^{13}+2a^{12}-26a^{11}-23a^{10}-38a^{9}-17a^{8}-13a^{7}+5a^{6}+41a^{5}+36a^{4}+58a^{3}+36a^{2}+36a-62$, $6a^{24}+a^{23}+5a^{22}+5a^{21}-4a^{20}+a^{19}-2a^{18}-7a^{17}-5a^{16}-14a^{15}-10a^{14}-8a^{13}-19a^{12}-12a^{11}-9a^{10}-10a^{9}-2a^{8}-5a^{7}+4a^{6}+21a^{5}+12a^{4}+18a^{3}+37a^{2}+30a+18$, $a^{24}-a^{23}-a^{22}+a^{21}+2a^{20}-a^{19}-2a^{18}+2a^{17}+2a^{16}-a^{15}-3a^{14}+a^{13}+4a^{12}-2a^{11}-4a^{10}+5a^{8}-7a^{6}+5a^{4}-7a^{2}-2a+6$, $2a^{24}+13a^{23}+14a^{22}-3a^{21}-10a^{20}+3a^{19}-24a^{17}-31a^{16}-14a^{15}-8a^{14}-28a^{13}-40a^{12}-17a^{11}+5a^{10}-17a^{9}-32a^{8}+16a^{7}+57a^{6}+35a^{5}+18a^{4}+59a^{3}+100a^{2}+68a+12$, $2a^{24}-6a^{22}+18a^{21}-a^{19}-13a^{18}-19a^{17}+16a^{16}+3a^{15}+14a^{14}-11a^{12}+17a^{11}-14a^{10}-7a^{9}-24a^{8}-12a^{7}+49a^{6}+22a^{5}+18a^{4}-38a^{3}-43a^{2}+19a-17$, $72a^{24}+52a^{23}+35a^{22}-29a^{20}-73a^{19}-107a^{18}-151a^{17}-178a^{16}-208a^{15}-215a^{14}-220a^{13}-197a^{12}-167a^{11}-107a^{10}-40a^{9}+54a^{8}+154a^{7}+268a^{6}+377a^{5}+479a^{4}+564a^{3}+615a^{2}+634a+378$
| 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: | \( 10786828262013908 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 10786828262013908 \cdot 1}{2\cdot\sqrt{5293954790280245387320429425326762678403863692388857}}\cr\approx \mathstrut & 0.561257387701037 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 3*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 - 3*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 - 3*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 - 3*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 $S_{25}$ are not computed |
| Character table for $S_{25}$ is not computed |
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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | $21{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(35846147\)
| $\Q_{35846147}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(76867751\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(192\!\cdots\!181\)
| $\Q_{19\!\cdots\!81}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |