Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 2*x - 3)
gp: K = bnfinit(y^33 - 2*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - 2*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - 2*x - 3)
\( x^{33} - 2x - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(239243498311888231191475443681693217502277009356802311496414548641\)
\(\medspace = 131009\cdot 18\!\cdots\!49\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(95.76\) | 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: | $131009^{1/2}1826160785227642613801154452607784331628185921248176167258849^{1/2}\approx 4.8912523786029305e+32$ | ||
| Ramified primes: |
\(131009\), \(18261\!\cdots\!58849\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{23924\!\cdots\!48641}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
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: | $16$ | 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: |
$a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+2$, $2a^{32}-a^{30}+2a^{29}-2a^{27}-a^{25}-4a^{24}+2a^{22}-2a^{21}+4a^{19}-3a^{17}+a^{16}-2a^{15}-5a^{14}-a^{13}-4a^{11}+2a^{10}+5a^{9}-a^{8}-a^{7}+4a^{6}-4a^{5}-7a^{4}-a^{3}-3a^{2}-8a-1$, $3a^{32}-3a^{31}-3a^{30}+5a^{28}-5a^{26}-3a^{25}+3a^{24}+5a^{23}-4a^{22}-6a^{21}+8a^{19}+4a^{18}-7a^{17}-7a^{16}+a^{15}+9a^{14}+a^{13}-8a^{12}-5a^{11}+6a^{10}+9a^{9}-6a^{8}-11a^{7}-a^{6}+12a^{5}+4a^{4}-15a^{3}-12a^{2}+6a+13$, $3a^{32}-4a^{31}+6a^{30}-2a^{29}-4a^{28}+4a^{27}-6a^{26}+a^{25}+5a^{24}-5a^{23}+5a^{22}+a^{21}-6a^{20}+5a^{19}-4a^{18}-2a^{17}+8a^{16}-6a^{15}+2a^{14}+4a^{13}-10a^{12}+5a^{11}-a^{10}-5a^{9}+13a^{8}-4a^{7}+8a^{5}-14a^{4}+a^{3}-a^{2}-10a+10$, $11a^{32}+a^{31}-3a^{30}-18a^{29}-7a^{28}-10a^{27}+12a^{26}+12a^{25}+19a^{24}+3a^{23}-4a^{22}-24a^{21}-13a^{20}-17a^{19}+16a^{18}+15a^{17}+30a^{16}+8a^{15}-5a^{14}-28a^{13}-26a^{12}-21a^{11}+14a^{10}+24a^{9}+42a^{8}+17a^{7}-6a^{6}-35a^{5}-46a^{4}-27a^{3}+5a^{2}+40a+34$, $a^{32}-a^{30}-4a^{29}-4a^{28}-6a^{27}-5a^{26}-5a^{25}-3a^{24}-2a^{23}+a^{22}+2a^{21}+5a^{20}+5a^{19}+6a^{18}+6a^{17}+3a^{16}+2a^{15}-4a^{14}-6a^{13}-10a^{12}-11a^{11}-11a^{10}-11a^{9}-9a^{8}-6a^{7}-a^{6}+5a^{5}+10a^{4}+12a^{3}+14a^{2}+10a+8$, $11a^{32}+12a^{31}+12a^{30}+14a^{29}+13a^{28}+14a^{27}+10a^{26}+12a^{25}+7a^{24}+6a^{23}+2a^{22}-a^{21}-6a^{20}-10a^{19}-12a^{18}-20a^{17}-20a^{16}-26a^{15}-26a^{14}-32a^{13}-28a^{12}-31a^{11}-28a^{10}-24a^{9}-20a^{8}-15a^{7}-9a^{6}+4a^{5}+7a^{4}+21a^{3}+28a^{2}+40a+22$, $a^{32}-4a^{31}+2a^{30}+a^{29}+4a^{27}+3a^{26}+a^{25}+10a^{24}+8a^{22}+5a^{21}+5a^{20}+6a^{19}+7a^{18}-2a^{17}+9a^{16}-2a^{15}+2a^{14}-7a^{12}-4a^{11}-3a^{10}-13a^{9}-6a^{8}-14a^{7}-14a^{6}-7a^{5}-16a^{4}-14a^{3}-7a^{2}-16a-4$, $7a^{32}-8a^{31}-15a^{30}-8a^{29}+13a^{28}+19a^{27}-a^{26}-21a^{25}-14a^{24}+9a^{23}+20a^{22}+14a^{21}-13a^{20}-31a^{19}-3a^{18}+28a^{17}+21a^{16}-6a^{15}-30a^{14}-25a^{13}+14a^{12}+42a^{11}+13a^{10}-36a^{9}-36a^{8}+a^{7}+39a^{6}+41a^{5}-14a^{4}-57a^{3}-29a^{2}+39a+46$, $a^{32}-a^{31}-3a^{30}+2a^{29}-7a^{28}+6a^{27}-6a^{26}+8a^{25}-8a^{24}+5a^{23}-7a^{22}+a^{21}-5a^{20}-a^{19}+4a^{18}-3a^{17}+5a^{16}-10a^{15}+6a^{14}-12a^{13}+4a^{12}-9a^{11}+7a^{10}-a^{9}-a^{7}-8a^{6}+a^{5}-15a^{4}+5a^{3}-7a^{2}+12a-10$, $23a^{32}+16a^{31}+7a^{30}+a^{29}-10a^{28}-22a^{27}-28a^{26}-29a^{25}-34a^{24}-24a^{23}-15a^{22}-a^{21}+12a^{20}+30a^{19}+39a^{18}+48a^{17}+44a^{16}+43a^{15}+23a^{14}+6a^{13}-17a^{12}-36a^{11}-63a^{10}-64a^{9}-70a^{8}-63a^{7}-43a^{6}-5a^{5}+12a^{4}+55a^{3}+86a^{2}+97a+55$, $2a^{32}-5a^{31}+4a^{30}-3a^{29}+4a^{28}+a^{25}-4a^{24}+3a^{23}-5a^{22}+5a^{21}-2a^{20}+3a^{19}+2a^{18}-5a^{17}+5a^{16}-10a^{15}+6a^{14}-4a^{13}+3a^{12}+6a^{11}-7a^{10}+8a^{9}-12a^{8}+4a^{7}-4a^{6}-2a^{5}+9a^{4}-5a^{3}+11a^{2}-7a-4$, $2a^{32}-2a^{31}-4a^{30}+a^{28}-2a^{27}-4a^{26}-5a^{25}-a^{24}+a^{23}-4a^{22}-6a^{21}-3a^{20}+a^{19}+a^{18}-3a^{17}-4a^{16}+7a^{14}+4a^{13}-4a^{12}+2a^{11}+8a^{10}+8a^{9}+6a^{8}-2a^{7}+3a^{6}+14a^{5}+7a^{4}-2a^{3}-2a^{2}+2a+4$, $2a^{32}+2a^{31}-2a^{30}+4a^{28}-2a^{27}-2a^{26}+2a^{25}+4a^{24}-3a^{23}-4a^{22}+3a^{21}-a^{20}-4a^{19}+6a^{17}-a^{16}-6a^{15}+3a^{14}+4a^{13}+6a^{10}-8a^{8}+a^{7}+4a^{6}-2a^{5}-5a^{4}+3a^{3}-a^{2}-10a-2$, $6a^{32}+3a^{31}-6a^{30}-4a^{29}+4a^{28}+2a^{27}-2a^{26}+3a^{25}+6a^{24}-a^{23}-7a^{22}-3a^{21}+7a^{20}+10a^{19}-8a^{17}-a^{16}+3a^{15}-5a^{14}-2a^{13}+12a^{12}+5a^{11}-16a^{10}-14a^{9}+5a^{8}+10a^{7}-6a^{5}-3a^{4}-8a^{2}-11a-2$, $4a^{32}+6a^{31}-a^{30}-6a^{29}+5a^{27}+6a^{26}-3a^{25}-7a^{24}-a^{23}+12a^{22}+3a^{21}-7a^{20}-6a^{19}+9a^{18}+7a^{17}+a^{16}-11a^{15}-3a^{14}+10a^{13}+14a^{12}-8a^{11}-13a^{10}+4a^{9}+18a^{8}+2a^{7}-11a^{6}-9a^{5}+11a^{4}+20a^{3}+3a^{2}-22a-14$
| 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: | \( 91796156039521290000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 91796156039521290000 \cdot 1}{2\cdot\sqrt{239243498311888231191475443681693217502277009356802311496414548641}}\cr\approx \mathstrut & 1.10734338886468 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 - 2*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^33 - 2*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^33 - 2*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^33 - 2*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 8683317618811886495518194401280000000 |
| The 10143 conjugacy class representatives for $S_{33}$ are not computed |
| Character table for $S_{33}$ 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 | ${\href{/padicField/2.10.0.1}{10} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $31{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $32{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $33$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(131009\)
| $\Q_{131009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{131009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(182\!\cdots\!849\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $31$ | $1$ | $31$ | $0$ | $C_{31}$ | $[\ ]^{31}$ |