Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x - 2)
gp: K = bnfinit(y^33 - y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - x - 2)
\( x^{33} - x - 2 \)
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: |
\(554523399760721082797165478848537512039766641116402126159872\)
\(\medspace = 2^{32}\cdot 31\cdot 491\cdot 35603\cdot 23\!\cdots\!39\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.63\) | 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: |
\(2\), \(31\), \(491\), \(35603\), \(23824\!\cdots\!38639\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{12911\!\cdots\!69057}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{17}+a+1$, $a^{2}+a+1$, $a^{22}+a^{11}+1$, $a^{32}-a^{31}+a^{30}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}-1$, $a^{25}-a^{17}+a^{9}-a-1$, $a^{29}-a^{25}+a^{21}-a^{17}+a^{13}-a^{9}+a^{5}-a-1$, $a^{31}-a^{29}+a^{27}-a^{25}+a^{23}-a^{21}+a^{19}-a^{17}+a^{15}-a^{13}+a^{11}-a^{9}+a^{7}-a^{5}+a^{3}-a-1$, $a^{32}-a^{31}-a^{30}-a^{27}+a^{25}-a^{24}-a^{21}-2a^{20}+a^{19}+a^{16}-2a^{14}-a^{13}+a^{12}-a^{11}+a^{9}-a^{7}+a^{6}-a^{5}-3a^{4}+a^{2}-1$, $2a^{32}-2a^{31}-a^{30}+2a^{29}-a^{27}+2a^{25}-2a^{24}-a^{23}+2a^{22}+2a^{21}-3a^{20}-a^{19}+2a^{18}+a^{17}-2a^{16}+a^{15}+2a^{14}-3a^{13}-a^{12}+2a^{11}+3a^{10}-4a^{9}+2a^{7}-2a^{5}+a^{4}+3a^{3}-4a^{2}+1$, $3a^{32}+7a^{31}+7a^{30}+7a^{29}+7a^{28}+4a^{27}+3a^{26}-a^{25}-5a^{24}-7a^{23}-10a^{22}-9a^{21}-9a^{20}-7a^{19}-3a^{18}+6a^{16}+9a^{15}+11a^{14}+14a^{13}+11a^{12}+10a^{11}+5a^{10}-a^{9}-5a^{8}-12a^{7}-15a^{6}-17a^{5}-17a^{4}-12a^{3}-9a^{2}-a+3$, $a^{31}+2a^{30}-a^{29}+a^{28}-2a^{27}+2a^{26}-2a^{25}-3a^{23}-a^{18}+3a^{17}+3a^{15}-2a^{14}+2a^{13}+2a^{11}-2a^{10}-2a^{9}-a^{8}-a^{7}-4a^{5}-a^{3}+4a^{2}+1$, $6a^{31}+3a^{30}-3a^{29}-4a^{28}-2a^{27}+2a^{26}+a^{25}+a^{24}+a^{22}-a^{21}-5a^{20}-3a^{19}+4a^{18}+7a^{17}+a^{16}-7a^{15}-7a^{14}+7a^{12}+5a^{11}-3a^{10}-4a^{9}-2a^{8}-3a^{6}+a^{5}+6a^{4}+5a^{3}-3a^{2}-13a-7$, $2a^{32}+a^{31}-a^{30}+2a^{29}+a^{27}+a^{26}+a^{25}+3a^{24}+2a^{22}+2a^{21}-2a^{20}+2a^{19}-a^{17}+a^{16}-a^{15}-3a^{13}-2a^{12}-a^{11}-6a^{10}-2a^{8}-4a^{7}-4a^{5}-a^{4}-4a^{3}-a^{2}+a-5$, $a^{32}-a^{31}+2a^{29}+4a^{28}+a^{27}-2a^{26}+a^{25}+4a^{24}+4a^{23}+a^{22}+2a^{21}+3a^{20}-2a^{18}+a^{17}+6a^{16}+a^{15}-4a^{14}-4a^{13}-a^{12}-4a^{10}-2a^{9}-3a^{7}-9a^{6}-7a^{5}+2a^{4}+3a^{3}-3a^{2}-6a-1$, $6a^{32}-16a^{31}-14a^{30}+10a^{29}+18a^{28}-4a^{27}-22a^{26}-8a^{25}+18a^{24}+15a^{23}-15a^{22}-24a^{21}+3a^{20}+25a^{19}+8a^{18}-25a^{17}-20a^{16}+17a^{15}+29a^{14}-3a^{13}-31a^{12}-10a^{11}+31a^{10}+27a^{9}-17a^{8}-33a^{7}+5a^{6}+40a^{5}+16a^{4}-33a^{3}-29a^{2}+22a+35$, $8a^{32}+2a^{31}-7a^{30}+4a^{29}-15a^{28}+13a^{27}-4a^{26}+9a^{25}+2a^{24}-13a^{23}+4a^{22}-13a^{21}+17a^{20}-2a^{19}+5a^{18}-16a^{16}+7a^{15}-10a^{14}+17a^{13}-a^{12}+4a^{11}-3a^{10}-20a^{9}+10a^{8}-6a^{7}+21a^{6}-a^{5}-5a^{4}-6a^{3}-19a^{2}+18a-11$
| 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: | \( 133008702440587730 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 133008702440587730 \cdot 1}{2\cdot\sqrt{554523399760721082797165478848537512039766641116402126159872}}\cr\approx \mathstrut & 1.05389662047802 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 - 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^33 - 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^33 - 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^33 - 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 8683317618811886495518194401280000000 |
| The 10143 conjugacy class representatives for $S_{33}$ are not computed |
| Character table for $S_{33}$ |
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 | $26{,}\,{\href{/padicField/3.7.0.1}{7} }$ | $17{,}\,16$ | $21{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $32{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | $19{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | $31{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,16$ | $23{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $[\ ]$ |
| Deg $32$ | $32$ | $1$ | $32$ | ||||
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 31.16.0.1 | $x^{16} + 28 x^{7} + 24 x^{6} + 26 x^{5} + 28 x^{4} + 11 x^{3} + 19 x^{2} + 27 x + 3$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
|
\(491\)
| $\Q_{491}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(35603\)
| $\Q_{35603}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{35603}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{35603}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(238\!\cdots\!639\)
| $\Q_{23\!\cdots\!39}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23\!\cdots\!39}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |