Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 + 5*x - 3)
gp: K = bnfinit(y^29 + 5*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + 5*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + 5*x - 3)
\( x^{29} + 5x - 3 \)
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: |
\(6173893524985612137541010925546999516718276445839672493539709\)
\(\medspace = 23\cdot 79\cdot 107\cdot 931421\cdot 23882194037539\cdot 14\!\cdots\!69\)
| 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: | $23^{1/2}79^{1/2}107^{1/2}931421^{1/2}23882194037539^{1/2}1427579104021681143732218630118134569^{1/2}\approx 2.4847320831400742e+30$ | ||
| Ramified primes: |
\(23\), \(79\), \(107\), \(931421\), \(23882194037539\), \(14275\!\cdots\!34569\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{61738\!\cdots\!39709}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$2a^{28}-3a^{27}+3a^{26}-2a^{25}+a^{24}-a^{23}+2a^{22}-3a^{21}+3a^{20}-2a^{19}+a^{18}-a^{17}+2a^{16}-3a^{15}+3a^{14}-2a^{13}+a^{12}-a^{11}+2a^{10}-3a^{9}+3a^{8}-2a^{7}+a^{6}-a^{5}+2a^{4}-3a^{3}+2a^{2}+3a-2$, $29a^{28}+17a^{27}+11a^{26}+7a^{25}+4a^{24}+a^{23}+2a^{22}+2a^{21}-a^{20}+2a^{19}-2a^{17}+2a^{16}+a^{15}-a^{14}-a^{13}+a^{12}-a^{11}+a^{10}+2a^{9}-5a^{8}+2a^{7}+a^{6}-2a^{5}+3a^{4}-4a^{3}+a+145$, $410a^{28}+235a^{27}+139a^{26}+97a^{25}+59a^{24}+23a^{23}+16a^{22}+16a^{21}-4a^{20}-9a^{19}+17a^{18}+19a^{17}-13a^{16}-10a^{15}+28a^{14}+18a^{13}-26a^{12}-19a^{11}+13a^{10}-6a^{9}-26a^{8}+12a^{7}+25a^{6}-21a^{5}-11a^{4}+56a^{3}+29a^{2}-62a+2017$, $5a^{28}+13a^{27}-6a^{26}+21a^{25}-5a^{24}+9a^{23}+3a^{22}-16a^{21}+18a^{20}-29a^{19}+4a^{18}-13a^{17}-20a^{16}+19a^{15}-38a^{14}+25a^{13}-12a^{12}+8a^{11}+31a^{10}-32a^{9}+64a^{8}-26a^{7}+29a^{6}+17a^{5}-48a^{4}+73a^{3}-79a^{2}+30a-2$, $6a^{28}+38a^{27}-54a^{26}+16a^{25}+40a^{24}-69a^{23}+30a^{22}+39a^{21}-85a^{20}+48a^{19}+36a^{18}-103a^{17}+72a^{16}+28a^{15}-123a^{14}+102a^{13}+13a^{12}-144a^{11}+142a^{10}-12a^{9}-157a^{8}+189a^{7}-41a^{6}-168a^{5}+247a^{4}-87a^{3}-176a^{2}+312a-125$, $20a^{28}-21a^{27}+6a^{26}-18a^{25}-28a^{24}+40a^{23}+25a^{22}-9a^{21}-16a^{20}-23a^{19}-2a^{18}-a^{17}+52a^{16}+34a^{15}-83a^{14}-5a^{13}+16a^{12}-44a^{11}+91a^{10}+42a^{9}-88a^{8}-28a^{7}-13a^{6}+32a^{5}+37a^{4}+41a^{3}+30a^{2}-188a+88$, $19a^{28}+5a^{27}+6a^{26}+5a^{25}+2a^{24}-3a^{23}+5a^{22}-2a^{21}+4a^{19}-3a^{18}-3a^{17}+7a^{16}-3a^{15}-2a^{14}+8a^{13}-12a^{12}+7a^{11}+2a^{10}-a^{9}-9a^{8}+15a^{7}-18a^{6}+13a^{5}-6a^{3}-9a^{2}+27a+61$, $9a^{28}-a^{27}+21a^{26}+7a^{25}-12a^{24}+7a^{23}-10a^{22}-14a^{21}+20a^{20}-6a^{19}-13a^{18}+21a^{17}-13a^{16}-3a^{15}+42a^{14}-17a^{13}-18a^{12}+22a^{11}-45a^{10}-10a^{9}+57a^{8}-30a^{7}-3a^{6}+47a^{5}-58a^{4}+8a^{3}+74a^{2}-58a+46$, $99a^{28}+87a^{27}+106a^{26}+87a^{25}+80a^{24}+32a^{23}-17a^{22}-58a^{21}-90a^{20}-83a^{19}-45a^{18}-14a^{17}+80a^{16}+74a^{15}+141a^{14}+61a^{13}+43a^{12}-50a^{11}-108a^{10}-132a^{9}-129a^{8}-74a^{7}+38a^{6}+69a^{5}+205a^{4}+124a^{3}+155a^{2}+9a+394$, $163a^{28}+32a^{27}-97a^{26}+253a^{25}-150a^{24}+31a^{23}+171a^{22}-306a^{21}+206a^{20}-83a^{19}-270a^{18}+326a^{17}-376a^{16}+35a^{15}+258a^{14}-498a^{13}+476a^{12}-69a^{11}-289a^{10}+727a^{9}-546a^{8}+234a^{7}+471a^{6}-850a^{5}+819a^{4}-319a^{3}-598a^{2}+1054a-413$, $6a^{28}+3a^{27}-15a^{26}+8a^{25}-13a^{24}+13a^{23}-31a^{22}+5a^{21}-11a^{20}-17a^{18}-30a^{17}+18a^{16}-28a^{15}+11a^{14}-64a^{13}+26a^{12}-17a^{11}+3a^{10}-38a^{9}-32a^{8}+48a^{7}-43a^{6}+32a^{5}-92a^{4}+82a^{3}-38a^{2}+28a-8$, $19a^{28}-16a^{27}+5a^{26}+a^{25}-7a^{24}+24a^{23}-37a^{22}+54a^{21}-65a^{20}+57a^{19}-49a^{18}+26a^{17}+7a^{16}-30a^{15}+56a^{14}-68a^{13}+73a^{12}-94a^{11}+98a^{10}-98a^{9}+91a^{8}-51a^{7}+3a^{6}+70a^{5}-155a^{4}+195a^{3}-218a^{2}+209a-74$, $16a^{28}+8a^{27}-8a^{26}-14a^{25}-4a^{24}+14a^{23}+19a^{22}-10a^{21}-27a^{20}-a^{19}+26a^{18}+13a^{17}-11a^{16}-24a^{15}-12a^{14}+27a^{13}+35a^{12}-19a^{11}-49a^{10}+a^{9}+41a^{8}+25a^{7}-17a^{6}-44a^{5}-27a^{4}+55a^{3}+61a^{2}-37a-5$, $59a^{28}+a^{27}-56a^{26}+24a^{25}+12a^{24}-71a^{23}+35a^{22}+87a^{21}-60a^{20}-28a^{19}+79a^{18}-57a^{17}-90a^{16}+108a^{15}+19a^{14}-105a^{13}+111a^{12}+86a^{11}-189a^{10}-24a^{9}+155a^{8}-132a^{7}-40a^{6}+257a^{5}-70a^{4}-224a^{3}+211a^{2}+39a-58$
| 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: | \( 21861542052281573000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 21861542052281573000 \cdot 1}{2\cdot\sqrt{6173893524985612137541010925546999516718276445839672493539709}}\cr\approx \mathstrut & 1.31498070976580 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 + 5*x - 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 + 5*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 + 5*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 + 5*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 $S_{29}$ |
| 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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | R | ${\href{/padicField/29.7.0.1}{7} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $24{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $28{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 23.20.0.1 | $x^{20} + x^{2} - 6 x + 20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
|
\(79\)
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 79.7.0.1 | $x^{7} + 4 x + 76$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 79.9.0.1 | $x^{9} + 57 x^{2} + 19 x + 76$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 79.11.0.1 | $x^{11} + 3 x + 76$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
|
\(107\)
| 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 107.13.0.1 | $x^{13} + 4 x + 105$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| 107.14.0.1 | $x^{14} + x^{2} - x + 6$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(931421\)
| $\Q_{931421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{931421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(23882194037539\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(142\!\cdots\!569\)
| $\Q_{14\!\cdots\!69}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ |