Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 2*x - 4)
gp: K = bnfinit(y^29 - 2*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 2*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 2*x - 4)
\( x^{29} - 2x - 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(46255321605876134336297695084858373134299667695653709938688\)
\(\medspace = 2^{54}\cdot 3\cdot 151\cdot 1208109059\cdot 46\!\cdots\!91\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.42\) | 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: |
\(2\), \(3\), \(151\), \(1208109059\), \(46917\!\cdots\!83491\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{25676\!\cdots\!70957}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{2}a^{28}$
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: | $14$ | 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: |
$a^{28}+a^{27}-a^{25}-a^{24}+a^{22}+a^{21}-a^{19}-a^{18}+a^{16}+a^{15}-a^{13}-a^{12}+a^{10}+a^{9}-a^{7}-a^{6}+a^{4}+a^{3}+1$, $2a^{28}+3a^{27}+2a^{26}+a^{25}+a^{24}+2a^{23}+a^{22}+a^{21}+3a^{20}+4a^{19}+4a^{18}+5a^{17}+7a^{16}+7a^{15}+5a^{14}+6a^{13}+6a^{12}+4a^{11}+a^{10}+5a^{9}+3a^{8}+3a^{7}+4a^{6}+11a^{5}+9a^{4}+11a^{3}+16a^{2}+18a+9$, $11a^{28}+16a^{27}+2a^{26}-8a^{25}-20a^{24}-13a^{23}+2a^{22}+18a^{21}+24a^{20}+8a^{19}-10a^{18}-30a^{17}-21a^{16}-3a^{15}+27a^{14}+34a^{13}+20a^{12}-13a^{11}-41a^{10}-37a^{9}-11a^{8}+36a^{7}+50a^{6}+39a^{5}-15a^{4}-54a^{3}-64a^{2}-24a+19$, $3a^{28}-5a^{26}+2a^{25}+2a^{24}-a^{23}+6a^{22}-2a^{21}-4a^{20}+5a^{19}+2a^{18}+a^{17}+8a^{16}-7a^{15}-3a^{14}+6a^{13}-a^{12}+3a^{11}+5a^{10}-15a^{9}-2a^{8}+3a^{7}-5a^{6}+6a^{5}-3a^{4}-20a^{3}+2a^{2}+a-9$, $14a^{28}+6a^{27}-6a^{26}-17a^{25}-5a^{24}+9a^{23}+21a^{22}+2a^{21}-14a^{20}-23a^{19}+a^{18}+18a^{17}+25a^{16}-5a^{15}-25a^{14}-25a^{13}+10a^{12}+30a^{11}+25a^{10}-14a^{9}-38a^{8}-24a^{7}+22a^{6}+46a^{5}+23a^{4}-29a^{3}-56a^{2}-20a+15$, $2a^{28}-a^{27}+6a^{26}-8a^{25}+6a^{24}-2a^{23}-2a^{21}+3a^{20}+4a^{19}-7a^{18}+3a^{17}-2a^{16}+7a^{15}-7a^{14}+4a^{13}+3a^{12}-5a^{11}-2a^{10}+3a^{9}+13a^{8}-18a^{7}+7a^{6}+a^{5}+4a^{4}-12a^{3}+16a^{2}+4a-23$, $19a^{28}+9a^{27}+12a^{26}+12a^{25}+20a^{24}+7a^{23}+21a^{22}+19a^{21}+10a^{20}+30a^{19}+12a^{18}+26a^{17}+22a^{16}+27a^{15}+21a^{14}+37a^{13}+22a^{12}+33a^{11}+44a^{10}+18a^{9}+54a^{8}+37a^{7}+39a^{6}+48a^{5}+60a^{4}+33a^{3}+70a^{2}+64a+3$, $3a^{28}-10a^{27}-4a^{26}-6a^{24}+11a^{23}+4a^{21}+14a^{20}-5a^{19}+13a^{18}+5a^{17}-13a^{16}+4a^{15}-19a^{14}-12a^{13}-18a^{11}+7a^{10}-4a^{9}+2a^{8}+31a^{7}+4a^{6}+27a^{5}+18a^{4}-18a^{3}+14a^{2}-18a-31$, $a^{28}-8a^{27}-6a^{26}+4a^{25}+6a^{24}+7a^{23}-4a^{22}-12a^{21}-4a^{20}+5a^{19}+12a^{18}+6a^{17}-12a^{16}-13a^{15}-2a^{14}+10a^{13}+18a^{12}-2a^{11}-17a^{10}-14a^{9}-4a^{8}+24a^{7}+17a^{6}-8a^{5}-21a^{4}-26a^{3}+14a^{2}+32a+11$, $9a^{28}+a^{27}-9a^{26}-18a^{25}-23a^{24}-25a^{23}-23a^{22}-17a^{21}-4a^{20}+12a^{19}+26a^{18}+35a^{17}+38a^{16}+36a^{15}+24a^{14}+6a^{13}-15a^{12}-37a^{11}-55a^{10}-67a^{9}-61a^{8}-41a^{7}-14a^{6}+18a^{5}+50a^{4}+83a^{3}+102a^{2}+100a+61$, $14a^{28}-16a^{27}-9a^{26}+8a^{25}+9a^{24}+7a^{23}-15a^{22}-18a^{21}+22a^{20}+19a^{19}-19a^{18}-19a^{17}+8a^{16}+17a^{15}+14a^{14}-22a^{13}-28a^{12}+27a^{11}+19a^{10}-15a^{9}+2a^{8}-19a^{7}-a^{6}+39a^{5}-15a^{4}-17a^{3}-2a^{2}-16a+25$, $14a^{28}-16a^{27}+8a^{26}+11a^{25}-21a^{24}+13a^{23}+4a^{22}-25a^{21}+21a^{20}+a^{19}-21a^{18}+30a^{17}-3a^{16}-20a^{15}+30a^{14}-18a^{13}-17a^{12}+32a^{11}-28a^{10}-2a^{9}+40a^{8}-34a^{7}+8a^{6}+34a^{5}-54a^{4}+12a^{3}+23a^{2}-61a+5$, $7a^{28}-3a^{27}-4a^{26}+9a^{25}-a^{24}-9a^{23}+9a^{22}+3a^{21}-11a^{20}+9a^{19}+6a^{18}-14a^{17}+6a^{16}+11a^{15}-13a^{14}+a^{13}+12a^{12}-15a^{11}-6a^{10}+15a^{9}-10a^{8}-13a^{7}+12a^{6}-4a^{5}-17a^{4}+15a^{3}+11a^{2}-22a-7$, $15a^{28}-16a^{27}+10a^{26}+2a^{25}-13a^{24}+16a^{23}-10a^{22}-2a^{21}+10a^{20}-13a^{19}+9a^{18}+2a^{17}-11a^{16}+19a^{15}-14a^{14}-5a^{13}+19a^{12}-28a^{11}+20a^{10}+a^{9}-17a^{8}+35a^{7}-35a^{6}+15a^{5}+10a^{4}-40a^{3}+45a^{2}-28a-27$
| 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: | \( 2408495413358637000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 2408495413358637000 \cdot 1}{2\cdot\sqrt{46255321605876134336297695084858373134299667695653709938688}}\cr\approx \mathstrut & 1.67372124263854 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - 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^29 - 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^29 - 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^29 - 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 8841761993739701954543616000000 |
| The 4565 conjugacy class representatives for $S_{29}$ |
| Character table for $S_{29}$ |
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 | R | $28{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $28{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $19{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.10.0.1}{10} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | $16{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| Deg $28$ | $28$ | $1$ | $54$ | ||||
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(151\)
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.3.0.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.23.0.1 | $x^{23} + 11 x + 145$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(1208109059\)
| $\Q_{1208109059}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1208109059}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(469\!\cdots\!491\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |