Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x - 4)
gp: K = bnfinit(y^25 - 2*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 2*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 2*x - 4)
\( x^{25} - 2x - 4 \)
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: |
\(6249999988811813392427491791373439564250917896192\)
\(\medspace = 2^{46}\cdot 13751\cdot 431377\cdot 76457289497\cdot 195834881900887\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(89.50\) | 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: |
\(2\), \(13751\), \(431377\), \(76457289497\), \(195834881900887\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{88817\!\cdots\!82553}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}$
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
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: |
$6a^{24}-3a^{23}-a^{22}+6a^{21}-9a^{20}+3a^{19}+3a^{18}-6a^{17}+8a^{16}-a^{15}-9a^{14}+9a^{13}-8a^{12}+a^{11}+11a^{10}-11a^{9}+3a^{8}+3a^{7}-14a^{6}+14a^{5}-10a^{3}+14a^{2}-15a-15$, $a^{24}-a^{22}+a^{21}-a^{19}-a^{18}-a^{17}-a^{16}-2a^{15}-a^{14}-2a^{13}-a^{12}-a^{10}-2a^{9}+a^{8}+4a^{7}+a^{6}+4a^{4}+6a^{3}+a^{2}+2a+5$, $2a^{24}-a^{23}-4a^{22}-8a^{21}-8a^{20}-a^{19}+3a^{18}+8a^{17}+5a^{16}+6a^{15}+3a^{14}+3a^{13}+3a^{12}-5a^{10}-18a^{9}-18a^{8}-12a^{7}+6a^{6}+14a^{5}+16a^{4}+14a^{3}+9a^{2}+10a+1$, $3a^{24}-4a^{23}+a^{22}+3a^{21}-6a^{20}+5a^{19}+2a^{18}-7a^{17}+8a^{16}-a^{15}-8a^{14}+10a^{13}-6a^{12}-6a^{11}+11a^{10}-9a^{9}+10a^{7}-8a^{6}+5a^{5}+5a^{4}-6a^{3}+5a^{2}-4a-9$, $a^{24}+a^{23}+a^{21}+a^{19}+a^{18}+2a^{17}+2a^{16}+4a^{15}+2a^{14}+6a^{13}+2a^{12}+7a^{11}+3a^{10}+7a^{9}+5a^{8}+8a^{7}+7a^{6}+10a^{5}+7a^{4}+11a^{3}+6a^{2}+10a+3$, $a^{24}+4a^{23}-10a^{22}+13a^{21}-20a^{20}+13a^{19}-15a^{18}+19a^{17}+2a^{16}+2a^{14}-23a^{13}+18a^{12}-21a^{11}+23a^{10}-17a^{9}+19a^{8}+6a^{7}-a^{6}+2a^{5}-42a^{4}+19a^{3}-20a^{2}+48a-13$, $2a^{22}+4a^{21}+4a^{20}+4a^{19}-6a^{18}-8a^{17}-5a^{16}+2a^{15}+2a^{14}+6a^{13}+9a^{12}+5a^{11}-3a^{10}-18a^{9}-8a^{8}-a^{7}+5a^{6}+9a^{5}+14a^{4}+14a^{3}-5a^{2}-23a-25$, $5a^{23}+3a^{22}-3a^{21}+4a^{20}-5a^{19}-a^{18}-11a^{17}+7a^{16}-13a^{15}+9a^{14}-5a^{13}+17a^{12}-8a^{11}+20a^{10}-9a^{9}+13a^{8}-18a^{7}+5a^{6}-21a^{5}+2a^{4}-21a^{3}+11a^{2}+13$, $7a^{24}+6a^{23}+7a^{22}+8a^{21}+8a^{20}+12a^{19}+10a^{18}+9a^{17}+9a^{16}+16a^{15}+15a^{14}+14a^{13}+11a^{12}+19a^{11}+19a^{10}+20a^{9}+20a^{8}+24a^{7}+18a^{6}+27a^{5}+36a^{4}+30a^{3}+20a^{2}+36a+33$, $6a^{23}-3a^{22}-6a^{21}+6a^{20}+5a^{19}-8a^{18}+a^{17}+11a^{16}-a^{15}-8a^{14}+10a^{13}+9a^{12}-11a^{11}-a^{10}+15a^{9}-2a^{8}-16a^{7}+10a^{6}+10a^{5}-20a^{4}-9a^{3}+18a^{2}-5a-27$, $3a^{24}-5a^{23}-5a^{22}-2a^{21}-8a^{20}-13a^{19}-12a^{18}-14a^{17}-13a^{16}-14a^{15}-16a^{14}-13a^{13}-2a^{12}-5a^{11}-4a^{10}+5a^{9}+17a^{8}+16a^{7}+17a^{6}+19a^{5}+34a^{4}+31a^{3}+12a^{2}+10a+21$, $3a^{24}+4a^{23}-9a^{22}+7a^{21}-2a^{20}-6a^{19}+12a^{18}-9a^{17}-2a^{16}+14a^{15}-12a^{14}+4a^{13}+8a^{12}-12a^{11}+14a^{10}-2a^{9}-13a^{8}+20a^{7}-9a^{6}-7a^{5}+14a^{4}-19a^{3}+9a^{2}+6a-35$
| 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: | \( 1436614179210074.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 1436614179210074.2 \cdot 1}{2\cdot\sqrt{6249999988811813392427491791373439564250917896192}}\cr\approx \mathstrut & 2.17549735980538 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x - 4)
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 - 2*x - 4, 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 - 2*x - 4);
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 - 2*x - 4);
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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $25$ | $17{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $25$ |
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 | $[\ ]$ |
| Deg $24$ | $24$ | $1$ | $46$ | ||||
|
\(13751\)
| $\Q_{13751}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13751}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(431377\)
| $\Q_{431377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(76457289497\)
| $\Q_{76457289497}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{76457289497}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(195834881900887\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |