Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x - 1)
gp: K = bnfinit(y^33 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - 3*x - 1)
\( x^{33} - 3x - 1 \)
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: | $[3, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-8124576120042424569795315330246821593959810515006765161304275935\)
\(\medspace = -\,3^{33}\cdot 5\cdot 40739\cdot 11781157\cdot 3137660461139\cdot 194099782564469344251877\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(86.43\) | 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: |
\(3\), \(5\), \(40739\), \(11781157\), \(3137660461139\), \(194099782564469344251877\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-43845\!\cdots\!01535}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $17$ | 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: |
$a^{32}-3$, $a+1$, $a^{21}-a^{10}-1$, $a^{17}-2a$, $a^{32}-a^{30}+a^{29}-a^{27}+a^{26}-a^{24}+a^{23}-a^{21}+a^{20}-a^{18}+a^{16}-a^{14}+a^{13}-a^{11}+a^{10}-a^{8}+a^{7}-a^{5}+a^{4}-a^{2}-a-1$, $9a^{32}+7a^{31}+a^{30}-5a^{29}-9a^{28}-13a^{27}-9a^{26}-5a^{25}+3a^{24}+10a^{23}+15a^{22}+14a^{21}+8a^{20}-9a^{18}-18a^{17}-17a^{16}-15a^{15}-4a^{14}+8a^{13}+19a^{12}+23a^{11}+20a^{10}+11a^{9}-4a^{8}-20a^{7}-26a^{6}-30a^{5}-18a^{4}-2a^{3}+17a^{2}+31a+10$, $3a^{32}-3a^{31}-a^{30}+a^{29}-3a^{28}+4a^{27}-4a^{26}+7a^{25}-4a^{24}+5a^{23}+a^{22}-2a^{21}+6a^{20}-6a^{19}+5a^{18}-7a^{17}+3a^{16}-6a^{15}+a^{13}-7a^{12}+9a^{11}-8a^{10}+9a^{9}-2a^{8}+5a^{7}+2a^{6}+6a^{4}-10a^{3}+10a^{2}-14a-5$, $2a^{32}-2a^{31}-a^{29}+2a^{28}-a^{25}+3a^{23}-a^{22}-a^{21}-a^{20}+2a^{19}-2a^{17}-a^{16}-2a^{15}+4a^{14}-a^{12}-2a^{11}+3a^{10}+3a^{9}-2a^{8}-2a^{6}+3a^{5}-2a^{3}-4a^{2}+a$, $5a^{32}-2a^{31}-5a^{30}-2a^{29}+6a^{28}+4a^{27}-3a^{26}-8a^{25}+6a^{23}+5a^{22}-4a^{21}-3a^{20}+2a^{19}+4a^{18}-5a^{17}-7a^{16}+a^{15}+11a^{14}+4a^{13}-8a^{12}-10a^{11}+6a^{10}+13a^{9}+4a^{8}-13a^{7}-9a^{6}+3a^{5}+8a^{4}-5a^{3}-6a^{2}+6a+1$, $a^{32}+a^{31}+4a^{30}+3a^{29}+a^{28}+2a^{27}-2a^{26}-5a^{25}-4a^{24}-5a^{23}-4a^{22}+3a^{21}+5a^{20}+7a^{19}+10a^{18}+4a^{17}-3a^{16}-5a^{15}-13a^{14}-12a^{13}-3a^{12}+9a^{10}+17a^{9}+9a^{8}+6a^{7}-13a^{5}-11a^{4}-8a^{3}-10a^{2}+a+2$, $a^{32}+2a^{31}+2a^{30}-2a^{28}-4a^{27}-3a^{26}-a^{25}+2a^{24}+2a^{23}+2a^{22}-a^{21}-a^{20}-a^{19}+a^{18}+2a^{17}+a^{16}-a^{15}-2a^{14}+3a^{12}+5a^{11}+3a^{10}-a^{9}-7a^{8}-8a^{7}-6a^{6}+3a^{4}+4a^{3}+2a^{2}+2a+2$, $2a^{32}+5a^{31}-6a^{30}+8a^{29}-4a^{28}+3a^{27}+5a^{26}-6a^{25}+9a^{24}-5a^{23}+5a^{22}+4a^{21}-6a^{20}+11a^{19}-6a^{18}+7a^{17}+2a^{16}-5a^{15}+13a^{14}-7a^{13}+9a^{12}+a^{11}-4a^{10}+14a^{9}-8a^{8}+12a^{7}-a^{6}-2a^{5}+16a^{4}-10a^{3}+14a^{2}-2a-4$, $2a^{32}+6a^{31}+4a^{30}+4a^{29}+3a^{28}+4a^{27}-2a^{26}-3a^{25}-5a^{24}-3a^{23}-8a^{22}-8a^{21}-5a^{20}+a^{19}-a^{18}+a^{17}+5a^{16}+13a^{15}+9a^{14}+6a^{13}+5a^{12}+9a^{11}-6a^{9}-12a^{8}-7a^{7}-12a^{6}-15a^{5}-13a^{4}-2a^{3}+5a+3$, $7a^{32}-7a^{30}+3a^{29}+5a^{28}-6a^{27}-4a^{26}+9a^{25}+2a^{24}-9a^{23}+8a^{21}-5a^{20}-5a^{19}+9a^{18}+4a^{17}-13a^{16}-a^{15}+12a^{14}-2a^{13}-10a^{12}+9a^{11}+7a^{10}-14a^{9}-5a^{8}+17a^{7}-15a^{5}+6a^{4}+14a^{3}-15a^{2}-10a-2$, $3a^{32}-4a^{31}+5a^{30}-4a^{29}+2a^{28}+a^{27}-4a^{26}+6a^{25}-6a^{24}+5a^{23}-3a^{22}+a^{21}-2a^{19}+3a^{18}-6a^{17}+9a^{16}-11a^{15}+12a^{14}-9a^{13}+4a^{12}+3a^{11}-9a^{10}+13a^{9}-13a^{8}+11a^{7}-7a^{6}+3a^{5}-2a^{4}-a^{3}+a^{2}-4a-2$, $4a^{32}+10a^{31}-9a^{30}-12a^{29}+6a^{28}+6a^{27}-5a^{26}-4a^{25}-4a^{24}-2a^{23}+11a^{22}+2a^{21}-18a^{20}+a^{19}+20a^{18}-a^{17}-11a^{16}+3a^{15}+5a^{14}+9a^{13}+10a^{12}-12a^{11}-12a^{10}+25a^{9}+19a^{8}-22a^{7}-13a^{6}+16a^{5}+8a^{4}+2a^{3}-8a^{2}-27a-7$, $2a^{32}-5a^{31}+3a^{30}+2a^{29}-3a^{28}-8a^{27}-2a^{26}+5a^{25}-3a^{24}-3a^{23}+2a^{22}+10a^{21}-a^{20}-6a^{19}+2a^{17}-2a^{16}-13a^{15}+a^{14}+9a^{13}+4a^{12}-5a^{11}+6a^{10}+15a^{9}-4a^{8}-10a^{7}-7a^{6}+6a^{5}-5a^{4}-13a^{3}+3a^{2}+17a+4$
| 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: | \( 10925780584629996000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{15}\cdot 10925780584629996000 \cdot 1}{2\cdot\sqrt{8124576120042424569795315330246821593959810515006765161304275935}}\cr\approx \mathstrut & 0.455313089060654 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x - 1)
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 - 3*x - 1, 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 - 3*x - 1);
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 - 3*x - 1);
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 | $15^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | R | $33$ | $27{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $21{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | $30{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/43.8.0.1}{8} }$ | $22{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.15.15.44 | $x^{15} + 6 x^{14} + 27 x^{13} + 294 x^{12} + 819 x^{11} + 1080 x^{10} + 144 x^{9} + 1863 x^{8} + 1296 x^{7} + 2781 x^{6} + 648 x^{5} + 3402 x^{4} - 1053 x^{3} + 972 x^{2} - 729 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
| 3.15.15.34 | $x^{15} - 18 x^{14} + 462 x^{13} + 3948 x^{12} + 46782 x^{11} + 173079 x^{10} + 158742 x^{9} + 124092 x^{8} + 152685 x^{7} + 212733 x^{6} + 149931 x^{5} + 90234 x^{4} + 37827 x^{3} + 12150 x^{2} + 2430 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.7.0.1 | $x^{7} + 3 x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 5.22.0.1 | $x^{22} + x^{12} + 3 x^{11} + 4 x^{9} + 3 x^{8} + 2 x^{6} + 2 x^{5} + 4 x^{3} + 3 x^{2} + 3 x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
|
\(40739\)
| $\Q_{40739}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{40739}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(11781157\)
| $\Q_{11781157}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11781157}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(3137660461139\)
| $\Q_{3137660461139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3137660461139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3137660461139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(194\!\cdots\!877\)
| $\Q_{19\!\cdots\!77}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |