Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 3*x - 5)
gp: K = bnfinit(y^29 - 3*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 3*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 3*x - 5)
\( x^{29} - 3x - 5 \)
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: |
\(95653760883874762898413710346737322910550120611307005808278237\)
\(\medspace = 42691141\cdot 481626857\cdot 2939004407579\cdot 35136939022453\cdot 45049443348110236823\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(137.17\) | 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: | $42691141^{1/2}481626857^{1/2}2939004407579^{1/2}35136939022453^{1/2}45049443348110236823^{1/2}\approx 9.780274069977526e+30$ | ||
| Ramified primes: |
\(42691141\), \(481626857\), \(2939004407579\), \(35136939022453\), \(45049443348110236823\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{95653\!\cdots\!78237}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$2a^{27}+a^{26}-2a^{24}-a^{23}+2a^{21}+a^{20}-2a^{18}-a^{17}+2a^{15}+a^{14}-4a^{13}-4a^{12}-a^{11}+4a^{10}+4a^{9}+a^{8}-4a^{7}-4a^{6}-a^{5}+4a^{4}+4a^{3}+a^{2}-4a-4$, $7a^{28}-8a^{27}-20a^{26}-9a^{25}+16a^{24}+23a^{23}+5a^{22}-17a^{21}-34a^{20}+3a^{19}+23a^{18}+36a^{17}-3a^{16}-35a^{15}-40a^{14}+16a^{13}+36a^{12}+51a^{11}-31a^{10}-50a^{9}-44a^{8}+36a^{7}+69a^{6}+41a^{5}-61a^{4}-75a^{3}-39a^{2}+84a+82$, $85a^{28}-89a^{27}+86a^{26}-83a^{25}+80a^{24}-75a^{23}+77a^{22}-72a^{21}+73a^{20}-64a^{19}+55a^{18}-43a^{17}+26a^{16}-20a^{15}+6a^{14}-8a^{13}-a^{12}+5a^{11}-15a^{10}+36a^{9}-46a^{8}+74a^{7}-76a^{6}+93a^{5}-93a^{4}+98a^{3}-116a^{2}+119a-412$, $2a^{28}+a^{27}+3a^{26}-13a^{25}+13a^{24}-4a^{23}-22a^{22}+20a^{21}-4a^{20}-13a^{19}+8a^{18}+13a^{17}-8a^{16}+3a^{15}+23a^{14}-25a^{13}+15a^{12}+12a^{11}-36a^{10}+6a^{9}+5a^{8}-11a^{7}-28a^{6}+20a^{5}+19a^{4}-29a^{3}+32a^{2}+16a-6$, $50a^{28}+87a^{27}-5a^{26}-103a^{25}-54a^{24}+95a^{23}+107a^{22}-53a^{21}-146a^{20}-23a^{19}+152a^{18}+112a^{17}-118a^{16}-193a^{15}+38a^{14}+229a^{13}+81a^{12}-211a^{11}-220a^{10}+131a^{9}+322a^{8}+14a^{7}-348a^{6}-205a^{5}+279a^{4}+393a^{3}-109a^{2}-519a-289$, $25a^{28}+30a^{27}+11a^{26}-84a^{25}+55a^{24}+53a^{23}-77a^{22}-15a^{21}+64a^{20}+38a^{19}-141a^{18}+72a^{17}+68a^{16}-56a^{15}-101a^{14}+124a^{13}+56a^{12}-178a^{11}+42a^{10}+133a^{9}-29a^{8}-206a^{7}+187a^{6}+101a^{5}-204a^{4}-57a^{3}+239a^{2}+2a-419$, $8a^{28}+8a^{27}-a^{26}-12a^{25}-20a^{24}-18a^{23}-3a^{22}+11a^{21}+14a^{20}-18a^{18}-34a^{17}-32a^{16}-8a^{15}+22a^{14}+37a^{13}+25a^{12}+3a^{11}-20a^{10}-17a^{9}+16a^{8}+66a^{7}+87a^{6}+66a^{5}+15a^{4}-40a^{3}-62a^{2}-34a+8$, $5a^{28}-19a^{27}+24a^{26}-34a^{25}+39a^{24}-36a^{23}+41a^{22}-30a^{21}+19a^{20}-12a^{19}-12a^{18}+18a^{17}-32a^{16}+49a^{15}-46a^{14}+58a^{13}-48a^{12}+36a^{11}-28a^{10}+5a^{9}+6a^{8}-31a^{7}+45a^{6}-62a^{5}+71a^{4}-61a^{3}+74a^{2}-49a+26$, $22a^{28}+25a^{27}+a^{26}-7a^{25}-23a^{24}-42a^{23}-18a^{22}+3a^{21}+15a^{20}+46a^{19}+39a^{18}+14a^{17}+8a^{16}-36a^{15}-64a^{14}-40a^{13}-38a^{12}+3a^{11}+75a^{10}+76a^{9}+76a^{8}+42a^{7}-67a^{6}-94a^{5}-91a^{4}-103a^{3}-8a^{2}+71a+41$, $21a^{28}+13a^{27}-15a^{26}-23a^{25}+3a^{24}+30a^{23}+16a^{22}-25a^{21}-29a^{20}+15a^{19}+37a^{18}+a^{17}-40a^{16}-24a^{15}+33a^{14}+49a^{13}-16a^{12}-73a^{11}-23a^{10}+66a^{9}+56a^{8}-40a^{7}-82a^{6}-7a^{5}+86a^{4}+52a^{3}-78a^{2}-96a-9$, $220a^{28}-26a^{27}-141a^{26}+26a^{25}+107a^{24}-137a^{23}-170a^{22}+281a^{21}+310a^{20}-277a^{19}-502a^{18}+97a^{17}+641a^{16}+99a^{15}-565a^{14}-185a^{13}+341a^{12}+108a^{11}-264a^{10}+124a^{9}+521a^{8}-338a^{7}-965a^{6}+204a^{5}+1322a^{4}+370a^{3}-1404a^{2}-936a+423$, $14a^{28}+58a^{27}+106a^{26}+153a^{25}+198a^{24}+242a^{23}+276a^{22}+307a^{21}+324a^{20}+327a^{19}+319a^{18}+280a^{17}+235a^{16}+148a^{15}+53a^{14}-76a^{13}-222a^{12}-372a^{11}-544a^{10}-683a^{9}-832a^{8}-923a^{7}-990a^{6}-994a^{5}-934a^{4}-822a^{3}-627a^{2}-393a-132$, $1948a^{28}-638a^{27}-811a^{26}+2272a^{25}-3584a^{24}+4619a^{23}-5263a^{22}+5416a^{21}-5042a^{20}+4153a^{19}-2799a^{18}+1081a^{17}+832a^{16}-2775a^{15}+4571a^{14}-5998a^{13}+6921a^{12}-7228a^{11}+6811a^{10}-5708a^{9}+3988a^{8}-1752a^{7}-781a^{6}+3372a^{5}-5789a^{4}+7786a^{3}-9097a^{2}+9600a-15053$, $184a^{28}-514a^{27}+262a^{26}+251a^{25}-411a^{24}+544a^{23}+112a^{22}-548a^{21}+479a^{20}-300a^{19}-641a^{18}+678a^{17}-309a^{16}-182a^{15}+1093a^{14}-447a^{13}-118a^{12}+767a^{11}-1206a^{10}-203a^{9}+641a^{8}-1196a^{7}+743a^{6}+1119a^{5}-970a^{4}+1184a^{3}+345a^{2}-1932a+257$
| 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: | \( 46446705639915990000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 46446705639915990000 \cdot 1}{2\cdot\sqrt{95653760883874762898413710346737322910550120611307005808278237}}\cr\approx \mathstrut & 0.709777205446492 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - 3*x - 5)
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 - 3*x - 5, 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 - 3*x - 5);
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 - 3*x - 5);
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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $16{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/37.8.0.1}{8} }$ | $15{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(42691141\)
| $\Q_{42691141}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(481626857\)
| $\Q_{481626857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
|
\(2939004407579\)
| $\Q_{2939004407579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2939004407579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(35136939022453\)
| $\Q_{35136939022453}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(45049443348110236823\)
| $\Q_{45049443348110236823}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |