Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 5*x - 2)
gp: K = bnfinit(y^29 - 5*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 5*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 5*x - 2)
\( x^{29} - 5x - 2 \)
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: | $[3, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-6173834783873138787922773540394418282985013768744056322523136\)
\(\medspace = -\,2^{28}\cdot 3\cdot 47\cdot 19231\cdot 1239599\cdot 17156593\cdot 53138329\cdot 7505402452445630661805687\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(124.80\) | 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: |
\(2\), \(3\), \(47\), \(19231\), \(1239599\), \(17156593\), \(53138329\), \(7505402452445630661805687\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-22999\!\cdots\!16531}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: |
$6a^{28}-2a^{27}+a^{24}-a^{23}-a^{22}+a^{21}-a^{20}-2a^{19}-a^{18}-2a^{16}-2a^{15}+a^{14}-2a^{12}+a^{11}+2a^{10}-a^{9}+3a^{7}+3a^{6}-a^{5}+4a^{4}+6a^{3}-25$, $12a^{28}-6a^{27}-6a^{26}-a^{25}-11a^{24}+15a^{23}+8a^{21}+4a^{20}-20a^{19}-9a^{17}+8a^{16}+17a^{15}+3a^{14}+5a^{13}-17a^{12}-21a^{11}+2a^{10}+4a^{9}+18a^{8}+27a^{7}-13a^{6}-8a^{5}-34a^{4}-18a^{3}+32a^{2}+a-5$, $65a^{28}-34a^{27}+20a^{26}-12a^{25}+6a^{24}-a^{23}-3a^{22}+5a^{21}-6a^{20}+5a^{19}-4a^{18}+4a^{17}-5a^{16}+7a^{15}-10a^{14}+13a^{13}-17a^{12}+20a^{11}-23a^{10}+24a^{9}-21a^{8}+15a^{7}-2a^{6}-10a^{5}+24a^{4}-35a^{3}+42a^{2}-45a-281$, $18a^{28}-7a^{27}-4a^{26}+14a^{25}-20a^{24}-2a^{23}+22a^{22}-4a^{21}-10a^{20}+6a^{19}-5a^{18}-20a^{17}+36a^{16}+6a^{15}-32a^{14}+13a^{13}-5a^{12}-14a^{11}+24a^{10}+29a^{9}-49a^{8}-7a^{7}+36a^{6}-35a^{5}+28a^{4}+30a^{3}-53a^{2}-27a-29$, $18a^{28}+10a^{27}+14a^{26}+26a^{25}+8a^{24}+15a^{23}+25a^{22}+23a^{21}+26a^{20}+12a^{19}+26a^{18}+41a^{17}+23a^{16}+29a^{15}+36a^{14}+32a^{13}+41a^{12}+52a^{11}+42a^{10}+28a^{9}+61a^{8}+73a^{7}+51a^{6}+52a^{5}+57a^{4}+96a^{3}+86a^{2}+53a+17$, $5a^{27}-3a^{26}-2a^{25}+4a^{24}-a^{23}-3a^{22}+3a^{21}+a^{20}-4a^{19}+3a^{18}-a^{17}+2a^{16}-5a^{15}+4a^{14}+4a^{13}-13a^{12}+11a^{11}+5a^{10}-22a^{9}+21a^{8}+a^{7}-25a^{6}+27a^{5}-2a^{4}-28a^{3}+34a^{2}-10a-19$, $58a^{28}-101a^{27}+15a^{26}-35a^{25}+96a^{24}+12a^{23}-22a^{22}-71a^{21}-65a^{20}+90a^{19}+26a^{18}+103a^{17}-166a^{16}-8a^{15}-120a^{14}+199a^{13}+14a^{12}+67a^{11}-198a^{10}-71a^{9}+48a^{8}+169a^{7}+176a^{6}-174a^{5}-101a^{4}-236a^{3}+325a^{2}+98a-27$, $a^{28}-18a^{27}+17a^{26}-24a^{25}+17a^{24}-14a^{23}-2a^{22}+9a^{21}-23a^{20}+28a^{19}-28a^{18}+29a^{17}-6a^{16}+9a^{15}+27a^{14}-22a^{13}+47a^{12}-32a^{11}+38a^{10}-11a^{9}+a^{8}+32a^{7}-37a^{6}+55a^{5}-58a^{4}+28a^{3}-36a^{2}-33a+7$, $4a^{28}+a^{27}-7a^{26}-3a^{25}-2a^{24}+2a^{23}-a^{22}-9a^{21}+2a^{20}+9a^{19}-4a^{18}-3a^{17}+4a^{16}+11a^{15}+7a^{14}-8a^{13}+6a^{12}+24a^{11}-3a^{10}-6a^{9}+12a^{8}+9a^{7}+4a^{6}-13a^{5}-15a^{4}+25a^{3}-7a^{2}-49a-19$, $a^{28}-a^{27}+a^{26}+2a^{25}-3a^{23}-2a^{22}+a^{21}+a^{20}-5a^{19}-3a^{18}+4a^{17}+4a^{16}-a^{15}+a^{14}+4a^{13}+3a^{12}-a^{11}-2a^{8}-8a^{7}-6a^{6}+4a^{5}+3a^{4}-9a^{3}+14a+5$, $4a^{28}+2a^{27}-6a^{25}-9a^{24}-6a^{23}+6a^{22}+4a^{21}-13a^{20}-11a^{19}-7a^{18}-2a^{17}+9a^{16}+4a^{15}-4a^{14}-15a^{13}-4a^{12}+28a^{11}+18a^{10}+3a^{9}+3a^{8}-6a^{7}+10a^{6}+33a^{5}+34a^{4}+a^{3}-32a^{2}-4a+3$, $4a^{28}-5a^{27}+16a^{26}-21a^{25}+a^{24}+32a^{23}-38a^{22}+5a^{21}+31a^{20}-32a^{19}+5a^{18}+11a^{17}-6a^{16}+6a^{15}-18a^{14}+16a^{13}+17a^{12}-42a^{11}+13a^{10}+40a^{9}-44a^{8}-16a^{7}+62a^{6}-25a^{5}-44a^{4}+57a^{3}-10a^{2}-21a-13$, $109a^{28}-93a^{27}+43a^{26}+4a^{25}-63a^{24}+122a^{23}-138a^{22}+145a^{21}-119a^{20}+46a^{19}+16a^{18}-97a^{17}+174a^{16}-187a^{15}+193a^{14}-154a^{13}+46a^{12}+37a^{11}-145a^{10}+246a^{9}-254a^{8}+261a^{7}-197a^{6}+40a^{5}+62a^{4}-209a^{3}+347a^{2}-341a-205$, $131a^{28}-48a^{27}+17a^{26}-9a^{25}-7a^{24}-7a^{23}-5a^{22}-6a^{21}+7a^{20}-2a^{19}+9a^{18}-a^{17}-4a^{16}-7a^{15}-16a^{14}-7a^{13}-12a^{12}+8a^{11}+3a^{10}+14a^{9}+9a^{8}-10a^{7}-3a^{6}-33a^{5}-16a^{4}-21a^{3}-9a^{2}+11a-647$, $a^{28}-16a^{27}-8a^{26}-7a^{25}-13a^{24}-a^{23}-6a^{22}+9a^{20}+a^{19}+17a^{18}+14a^{17}+17a^{16}+31a^{15}+17a^{14}+31a^{13}+25a^{12}+17a^{11}+31a^{10}+7a^{9}+12a^{8}+2a^{7}-22a^{6}-8a^{5}-40a^{4}-41a^{3}-46a^{2}-78a-59$
| 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: | \( 48016386518967206000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{13}\cdot 48016386518967206000 \cdot 1}{2\cdot\sqrt{6173834783873138787922773540394418282985013768744056322523136}}\cr\approx \mathstrut & 1.83869735949242 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - 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^29 - 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^29 - 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^29 - 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 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 | R | R | ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| 2.12.12.32 | $x^{12} - 4 x^{10} - 4 x^{9} + 26 x^{8} + 40 x^{7} - 4 x^{6} - 40 x^{5} + 28 x^{4} + 72 x^{3} + 24 x^{2} - 16 x + 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
| 2.12.12.31 | $x^{12} + 2 x^{10} + 2 x^{9} + 6 x^{8} + 12 x^{7} + 32 x^{6} + 48 x^{5} + 76 x^{4} + 48 x^{3} + 40 x^{2} + 8 x + 8$ | $4$ | $3$ | $12$ | 12T205 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
|
\(3\)
| 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 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
|
\(19231\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
|
\(1239599\)
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(17156593\)
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(53138329\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
|
\(750\!\cdots\!687\)
| $\Q_{75\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |