Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 4)
gp: K = bnfinit(y^25 + 5*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 + 5*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 + 5*x - 4)
\( x^{25} + 5x - 4 \)
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: |
\(422484236016954220667207680000000000000000000000000\)
\(\medspace = 2^{48}\cdot 5^{25}\cdot 23\cdot 29\cdot 35541523\cdot 212450981\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.93\) | 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: |
\(2\), \(5\), \(23\), \(29\), \(35541523\), \(212450981\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{25182\!\cdots\!50105}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$10a^{24}-16a^{23}-29a^{22}+19a^{21}-4a^{20}-37a^{19}+21a^{18}+12a^{17}-47a^{16}+18a^{15}+33a^{14}-54a^{13}+6a^{12}+59a^{11}-57a^{10}-18a^{9}+85a^{8}-52a^{7}-55a^{6}+106a^{5}-30a^{4}-105a^{3}+123a^{2}+10a-105$, $a^{24}-3a^{23}+7a^{22}-3a^{21}-3a^{20}+7a^{19}-8a^{18}+8a^{16}-11a^{15}+6a^{14}+5a^{13}-15a^{12}+11a^{11}-3a^{10}-17a^{9}+19a^{8}-11a^{7}-10a^{6}+26a^{5}-23a^{4}+25a^{2}-40a+21$, $53a^{24}+45a^{23}+37a^{22}+32a^{21}+27a^{20}+19a^{19}+13a^{18}+12a^{17}+6a^{16}+a^{14}-3a^{13}-4a^{12}+4a^{11}+4a^{10}+a^{9}+7a^{8}+11a^{7}+7a^{6}+9a^{5}+10a^{4}-6a^{3}-5a^{2}+4a+253$, $a^{24}+2a^{22}+2a^{21}+2a^{20}+4a^{19}+2a^{18}+6a^{17}+a^{16}+8a^{15}+10a^{13}-3a^{12}+12a^{11}-5a^{10}+12a^{9}-7a^{8}+13a^{7}-10a^{6}+12a^{5}-11a^{4}+11a^{3}-12a^{2}+6a-5$, $14a^{24}+25a^{23}+34a^{22}+36a^{21}+25a^{20}+15a^{19}+11a^{18}+17a^{17}+32a^{16}+36a^{15}+27a^{14}+7a^{13}-10a^{12}-13a^{11}+4a^{10}+17a^{9}+16a^{8}-a^{7}-38a^{6}-48a^{5}-40a^{4}-13a^{3}+11a^{2}+33$, $12a^{24}-14a^{23}-24a^{22}-2a^{21}+26a^{20}+22a^{19}-13a^{18}-35a^{17}-12a^{16}+31a^{15}+37a^{14}-7a^{13}-48a^{12}-27a^{11}+34a^{10}+57a^{9}+3a^{8}-63a^{7}-51a^{6}+33a^{5}+85a^{4}+26a^{3}-77a^{2}-89a+81$, $51a^{24}+36a^{23}-33a^{22}-68a^{21}-11a^{20}+67a^{19}+61a^{18}-28a^{17}-93a^{16}-41a^{15}+77a^{14}+106a^{13}-13a^{12}-130a^{11}-79a^{10}+81a^{9}+154a^{8}+30a^{7}-163a^{6}-155a^{5}+76a^{4}+229a^{3}+84a^{2}-190a+15$, $a^{24}-9a^{23}+12a^{22}-6a^{21}-10a^{20}+18a^{19}-10a^{18}-6a^{17}+17a^{16}-20a^{15}+9a^{14}+16a^{13}-34a^{12}+22a^{11}+11a^{10}-34a^{9}+33a^{8}-12a^{7}-26a^{6}+58a^{5}-42a^{4}-21a^{3}+71a^{2}-64a+19$, $4a^{24}+8a^{23}+5a^{22}-3a^{21}+13a^{20}-4a^{19}-a^{18}+3a^{17}+5a^{16}-13a^{15}+14a^{14}-2a^{13}-a^{12}-2a^{11}+10a^{10}-16a^{9}+4a^{7}-7a^{6}-11a^{5}+18a^{4}+2a^{3}-29a^{2}+33a+9$, $8a^{24}-28a^{23}+23a^{22}+5a^{21}-28a^{20}+37a^{19}-9a^{18}-13a^{17}+58a^{16}-32a^{15}+a^{14}+62a^{13}-52a^{12}+41a^{11}+52a^{10}-86a^{9}+78a^{8}+25a^{7}-87a^{6}+127a^{5}-46a^{4}-87a^{3}+172a^{2}-113a+9$, $a^{23}-2a^{22}+5a^{21}-9a^{20}+15a^{19}-21a^{18}+27a^{17}-30a^{16}+29a^{15}-23a^{14}+13a^{13}-13a^{11}+23a^{10}-29a^{9}+30a^{8}-27a^{7}+21a^{6}-15a^{5}+9a^{4}-5a^{3}+2a^{2}-a+1$, $25a^{24}-26a^{23}+3a^{22}+22a^{21}-18a^{20}-4a^{19}+11a^{18}+20a^{17}-54a^{16}+44a^{15}+21a^{14}-84a^{13}+80a^{12}-12a^{11}-44a^{10}+27a^{9}+31a^{8}-41a^{7}-37a^{6}+114a^{5}-80a^{4}-61a^{3}+157a^{2}-80a+3$
| 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: | \( 8383885813460554.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 8383885813460554.0 \cdot 1}{2\cdot\sqrt{422484236016954220667207680000000000000000000000000}}\cr\approx \mathstrut & 1.54418212372167 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 + 5*x - 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 + 5*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 + 5*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 + 5*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 $S_{25}$ 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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | R | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.16.64 | $x^{8} + 2 x^{4} + 4 x + 2$ | $8$ | $1$ | $16$ | $V_4^2:(S_3\times C_2)$ | $[4/3, 4/3, 2, 7/3, 7/3]_{3}^{2}$ | |
| Deg $16$ | $8$ | $2$ | $32$ | ||||
|
\(5\)
| Deg $25$ | $25$ | $1$ | $25$ | |||
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.16.0.1 | $x^{16} + 19 x^{7} + 19 x^{6} + 16 x^{5} + 13 x^{4} + x^{3} + 14 x^{2} + 17 x + 5$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.22.0.1 | $x^{22} - x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
|
\(35541523\)
| $\Q_{35541523}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(212450981\)
| $\Q_{212450981}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |