Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 4*x - 3)
gp: K = bnfinit(y^29 + 4*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 4*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 4*x - 3)
\( x^{29} + 4x - 3 \)
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: |
\(9612287017893622166734097540299289246929146924131209511293\)
\(\medspace = 3744524429\cdot 8675517281441\cdot 29\!\cdots\!37\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(99.86\) | 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: | $3744524429^{1/2}8675517281441^{1/2}295892994260421814018306476982622737^{1/2}\approx 9.804227158676823e+28$ | ||
| Ramified primes: |
\(3744524429\), \(8675517281441\), \(29589\!\cdots\!22737\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{96122\!\cdots\!11293}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
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: | $14$ | 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: |
$a^{18}+a^{17}-a^{15}-a^{14}+a^{12}+a^{11}-a^{8}-a^{7}+a^{5}+a^{4}-a^{2}-a+1$, $a^{26}-a^{23}+a^{20}-a^{17}+a^{14}-a^{11}-a^{10}+a^{9}+a^{8}+a^{7}-a^{6}-a^{5}-2a^{4}+2a^{3}+a^{2}+2a-2$, $a^{28}-a^{26}+a^{25}-a^{24}+a^{23}-2a^{22}+2a^{21}-a^{20}+a^{19}-a^{18}+a^{17}+a^{16}-3a^{15}+3a^{14}-3a^{13}+3a^{12}-4a^{11}+3a^{10}-a^{8}+a^{7}-a^{6}+4a^{5}-7a^{4}+6a^{3}-5a^{2}+5a-2$, $12a^{28}+3a^{27}-a^{26}+11a^{25}-3a^{24}-6a^{23}+7a^{22}-11a^{21}-10a^{20}+7a^{19}-14a^{18}-8a^{17}+12a^{16}-12a^{15}+a^{14}+20a^{13}-11a^{12}+8a^{11}+24a^{10}-16a^{9}+10a^{8}+20a^{7}-30a^{6}+8a^{5}+14a^{4}-42a^{3}+9a^{2}+10a+1$, $6a^{28}+11a^{27}+15a^{26}+12a^{25}+12a^{24}+12a^{23}+15a^{22}+20a^{21}+15a^{20}+12a^{19}+13a^{18}+14a^{17}+19a^{16}+13a^{15}+6a^{14}+8a^{13}+8a^{12}+13a^{11}+7a^{10}-6a^{9}-a^{8}+2a^{6}-19a^{4}-13a^{3}-5a^{2}-9a+19$, $a^{28}-a^{26}-2a^{25}-2a^{24}-2a^{23}-3a^{22}-5a^{21}-8a^{20}-10a^{19}-9a^{18}-8a^{17}-9a^{16}-8a^{15}-5a^{14}-5a^{13}-6a^{12}-4a^{11}-3a^{10}-6a^{9}-8a^{8}-7a^{7}-5a^{6}-a^{5}+3a^{4}+3a^{3}+3a^{2}+6a+10$, $2a^{28}+2a^{27}-2a^{26}+5a^{25}-5a^{24}+a^{23}-3a^{22}-5a^{21}+3a^{20}-7a^{19}+4a^{18}-5a^{17}-4a^{16}-13a^{14}+6a^{13}-10a^{12}+a^{11}+3a^{10}-13a^{9}+10a^{8}-18a^{7}+5a^{6}-3a^{5}-8a^{4}+17a^{3}-19a^{2}+20a-7$, $a^{28}-3a^{27}+a^{26}-a^{24}+a^{23}-5a^{22}+4a^{21}-8a^{20}+8a^{19}-11a^{18}+13a^{17}-13a^{16}+19a^{15}-18a^{14}+22a^{13}-23a^{12}+26a^{11}-26a^{10}+29a^{9}-30a^{8}+32a^{7}-29a^{6}+34a^{5}-28a^{4}+31a^{3}-26a^{2}+27a-16$, $4a^{28}+6a^{27}-3a^{26}-15a^{25}-15a^{24}+6a^{23}+28a^{22}+18a^{21}-17a^{20}-30a^{19}-4a^{18}+19a^{17}+8a^{16}-10a^{15}-5a^{14}+13a^{13}+19a^{12}-32a^{10}-38a^{9}+8a^{8}+59a^{7}+37a^{6}-35a^{5}-52a^{4}+2a^{3}+35a^{2}+4a-13$, $18a^{28}+27a^{27}+28a^{26}+28a^{25}+21a^{24}+16a^{23}+10a^{22}+a^{21}-13a^{20}-22a^{19}-34a^{18}-36a^{17}-28a^{16}-25a^{15}-12a^{14}-10a^{13}+4a^{12}+16a^{11}+29a^{10}+43a^{9}+42a^{8}+32a^{7}+23a^{6}+14a^{5}-2a^{4}+2a^{3}-27a^{2}-36a+22$, $437a^{28}+321a^{27}+247a^{26}+179a^{25}+141a^{24}+97a^{23}+81a^{22}+56a^{21}+46a^{20}+26a^{19}+28a^{18}+20a^{17}+13a^{16}+5a^{15}+10a^{14}+13a^{13}-2a^{12}+2a^{11}+a^{10}+15a^{9}-13a^{8}+7a^{7}-7a^{6}+20a^{5}-22a^{4}+16a^{3}-16a^{2}+24a+1720$, $5a^{27}+2a^{26}-6a^{24}-2a^{23}+12a^{21}+11a^{20}+11a^{19}-6a^{18}-13a^{17}-26a^{16}-15a^{15}-4a^{14}+22a^{13}+26a^{12}+29a^{11}+6a^{10}-12a^{9}-32a^{8}-28a^{7}-16a^{6}+9a^{5}+26a^{4}+24a^{3}+12a^{2}-9a-11$, $8a^{28}+12a^{27}-17a^{26}+20a^{25}-7a^{24}-10a^{23}+25a^{22}-26a^{21}+12a^{20}+14a^{19}-30a^{18}+33a^{17}-17a^{16}-13a^{15}+42a^{14}-44a^{13}+21a^{12}+16a^{11}-51a^{10}+59a^{9}-34a^{8}-18a^{7}+66a^{6}-79a^{5}+43a^{4}+18a^{3}-77a^{2}+105a-32$, $4a^{28}+7a^{27}-6a^{26}+11a^{24}-9a^{23}-9a^{22}+15a^{21}+4a^{20}-9a^{19}+3a^{16}-8a^{14}+13a^{13}+4a^{12}-28a^{11}+9a^{10}+24a^{9}-21a^{8}+21a^{6}-20a^{5}-8a^{4}+15a^{3}+4a^{2}+8a-5$
| 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: | \( 632109057751709300 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 632109057751709300 \cdot 1}{2\cdot\sqrt{9612287017893622166734097540299289246929146924131209511293}}\cr\approx \mathstrut & 0.963600031458069 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 4*x - 3)
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 + 4*x - 3, 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 + 4*x - 3);
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 + 4*x - 3);
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 $S_{29}$ are not computed |
| Character table for $S_{29}$ 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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $22{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $22{,}\,{\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | $27{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $29$ | $17{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3744524429\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(8675517281441\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(295\!\cdots\!737\)
| $\Q_{29\!\cdots\!37}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29\!\cdots\!37}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29\!\cdots\!37}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |