Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 2)
gp: K = bnfinit(y^25 - 5*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x - 2)
\( x^{25} - 5x - 2 \)
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: | $[3, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-397484234526838101282442055000000000000000000000000\)
\(\medspace = -\,2^{24}\cdot 5^{25}\cdot 11\cdot 7226986082306147296044401\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.68\) | 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: |
\(2\), \(5\), \(11\), \(7226986082306147296044401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-39748\!\cdots\!42055}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $13$ | 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: |
$a^{24}-a^{21}-a^{19}+a^{16}-a^{13}-a^{12}-a^{11}-a^{10}+a^{9}+2a^{8}+a^{7}-a^{6}-a^{5}-a^{4}-2a^{3}-a^{2}-1$, $2a^{24}-a^{23}+a^{22}-a^{21}+2a^{20}-2a^{17}+a^{16}-a^{15}+a^{14}-a^{13}-4a^{11}+a^{10}-a^{9}+2a^{8}-3a^{7}+2a^{6}-2a^{5}+a^{4}-a^{3}+6a^{2}-7$, $36a^{24}-34a^{23}-6a^{22}-13a^{21}-3a^{20}+4a^{18}+7a^{17}+13a^{16}+22a^{15}+31a^{14}+36a^{13}+35a^{12}+30a^{11}+26a^{10}+25a^{9}+23a^{8}+14a^{7}-5a^{6}-29a^{5}-49a^{4}-60a^{3}-65a^{2}-72a-265$, $1098a^{24}-439a^{23}+177a^{22}-71a^{21}+28a^{20}-12a^{19}+5a^{18}-a^{17}+2a^{16}-a^{15}-2a^{13}+a^{12}+a^{11}+a^{10}+a^{9}-4a^{8}+a^{7}-3a^{6}+5a^{5}+a^{3}-2a^{2}-4a-5487$, $3a^{24}-18a^{23}+15a^{22}+4a^{21}-22a^{20}+17a^{19}+6a^{18}-27a^{17}+19a^{16}+9a^{15}-33a^{14}+21a^{13}+13a^{12}-39a^{11}+25a^{10}+18a^{9}-47a^{8}+30a^{7}+24a^{6}-58a^{5}+35a^{4}+33a^{3}-70a^{2}+38a+27$, $a^{24}+12a^{23}-5a^{22}-5a^{21}+14a^{20}+7a^{19}-13a^{18}+8a^{17}+17a^{16}-10a^{15}-11a^{14}+21a^{13}-a^{12}-27a^{11}+4a^{10}+18a^{9}-31a^{8}-21a^{7}+23a^{6}-8a^{5}-45a^{4}+18a^{3}+27a^{2}-37a-17$, $10a^{24}+4a^{23}-10a^{22}-4a^{21}+10a^{20}-a^{19}-10a^{18}+18a^{16}+a^{15}-24a^{14}-5a^{13}+22a^{12}+12a^{11}-21a^{10}-12a^{9}+22a^{8}+a^{7}-25a^{6}-4a^{5}+43a^{4}+11a^{3}-56a^{2}-24a-1$, $8a^{24}-12a^{23}+7a^{22}+4a^{21}-11a^{20}+6a^{19}+4a^{18}-8a^{17}+3a^{16}+7a^{15}-10a^{14}+10a^{12}-9a^{11}-a^{10}+13a^{9}-11a^{8}-10a^{7}+25a^{6}-15a^{5}-13a^{4}+40a^{3}-35a^{2}-8a+5$, $3a^{24}-17a^{23}+18a^{22}-17a^{21}+22a^{20}-20a^{19}+18a^{17}-17a^{16}+15a^{15}-24a^{14}+23a^{13}+5a^{12}-26a^{11}+20a^{10}-12a^{9}+24a^{8}-18a^{7}-23a^{6}+52a^{5}-39a^{4}+27a^{3}-43a^{2}+27a+21$, $a^{24}+3a^{23}-17a^{22}+13a^{21}-4a^{20}+7a^{19}-2a^{18}-16a^{17}+17a^{16}-8a^{15}+10a^{14}-13a^{13}-15a^{12}+26a^{11}-8a^{10}+23a^{9}-28a^{8}-7a^{7}+24a^{6}-11a^{5}+31a^{4}-49a^{3}+20a^{2}+21a-1$, $2a^{24}+a^{23}+6a^{22}+5a^{21}+a^{20}+8a^{19}+4a^{18}+2a^{17}+8a^{16}+9a^{15}+2a^{14}+13a^{13}+6a^{12}+4a^{11}+13a^{10}+15a^{9}+4a^{8}+19a^{7}+13a^{6}+6a^{5}+20a^{4}+23a^{3}+10a^{2}+28a+11$, $18a^{24}+2a^{23}+9a^{22}+17a^{21}-20a^{20}+13a^{19}-28a^{18}+6a^{17}+10a^{16}+2a^{15}+45a^{14}-15a^{13}+21a^{12}-25a^{11}-26a^{10}+16a^{9}-24a^{8}+71a^{7}+8a^{6}+38a^{5}+10a^{4}-61a^{3}+14a^{2}-67a-33$, $6a^{24}+8a^{23}-5a^{22}-19a^{21}+6a^{20}+20a^{19}-3a^{18}-14a^{17}-6a^{16}+5a^{15}+14a^{14}+12a^{13}-20a^{12}-25a^{11}+23a^{10}+31a^{9}-4a^{8}-36a^{7}-16a^{6}+36a^{5}+29a^{4}-17a^{3}-40a^{2}-15a-1$
| 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: | \( 24297329895974496 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 24297329895974496 \cdot 1}{2\cdot\sqrt{397484234526838101282442055000000000000000000000000}}\cr\approx \mathstrut & 2.93722448370821 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x - 2)
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 - 2, 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 - 2);
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 - 2);
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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/29.9.0.1}{9} }$ | $22{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.12.0.1}{12} }$ | $25$ | $25$ | $18{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.8.13 | $x^{8} + 2 x + 2$ | $8$ | $1$ | $8$ | $C_2^3:(C_7: C_3)$ | $[8/7, 8/7, 8/7]_{7}^{3}$ | |
| Deg $16$ | $8$ | $2$ | $16$ | ||||
|
\(5\)
| Deg $25$ | $25$ | $1$ | $25$ | |||
|
\(11\)
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.8.0.1 | $x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 11.15.0.1 | $x^{15} + 10 x^{6} + 7 x^{5} + 5 x^{3} + 9$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(722\!\cdots\!401\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |