Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x + 3)
gp: K = bnfinit(y^30 - 3*y + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 3*x + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 3*x + 3)
\( x^{30} - 3x + 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-13601722282800618439068316459204349535511987442947558142619\)
\(\medspace = -\,3^{30}\cdot 13\cdot 50\!\cdots\!87\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(86.65\) | 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: |
\(3\), \(13\), \(50817\!\cdots\!85887\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-66062\!\cdots\!16531}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
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
$C_{2}$, which has order $2$ (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: |
$a-1$, $a^{26}+a^{23}+a^{19}-a^{17}+a^{16}+a^{12}+a^{11}-2a^{10}+a^{9}-a^{7}+a^{5}+2a^{4}-3a^{3}+a^{2}+a-2$, $a^{25}+a^{24}+a^{23}+a^{21}+a^{20}+a^{19}-a^{17}+a^{16}-2a^{13}-a^{12}-a^{9}-2a^{8}+a^{5}-a^{4}+2a+1$, $a^{21}+a^{18}-a^{16}+a^{15}-a^{13}+a^{12}-a^{10}+a^{9}-a^{7}+a^{6}-a^{4}+a^{3}+1$, $a^{29}+3a^{28}+a^{27}+a^{25}-a^{24}-a^{23}+3a^{22}+2a^{21}+2a^{19}-3a^{17}+a^{16}+3a^{15}+2a^{13}+3a^{12}-4a^{11}-3a^{10}+4a^{9}-a^{7}+6a^{6}-6a^{4}+a^{3}+a^{2}-3a+2$, $3a^{29}+a^{28}-5a^{27}+4a^{26}-4a^{25}+6a^{24}-9a^{23}+5a^{22}-a^{21}+7a^{20}+a^{19}-6a^{18}+3a^{17}-7a^{16}+12a^{15}-14a^{14}+8a^{13}-8a^{12}+14a^{11}-4a^{9}-12a^{7}+20a^{6}-19a^{5}+15a^{4}-22a^{3}+25a^{2}-5a-5$, $6a^{28}+4a^{27}+4a^{25}+a^{24}+2a^{23}+8a^{22}-a^{20}+6a^{19}+2a^{17}+5a^{16}-5a^{15}+3a^{14}+8a^{13}-5a^{12}+a^{11}+3a^{10}-5a^{9}+9a^{8}+3a^{7}-11a^{6}+7a^{5}+3a^{4}-7a^{3}+9a^{2}-6a-7$, $a^{29}+3a^{28}+a^{26}-4a^{25}+a^{24}-4a^{23}+2a^{22}-3a^{21}+6a^{20}-2a^{19}+5a^{18}-4a^{17}+3a^{16}-8a^{15}+2a^{14}-4a^{13}+6a^{12}-3a^{11}+7a^{10}-2a^{9}+3a^{8}-6a^{7}+2a^{6}-5a^{5}-a^{4}+5a^{2}+5a-5$, $57a^{29}+41a^{28}+24a^{27}+5a^{26}-14a^{25}-32a^{24}-46a^{23}-59a^{22}-67a^{21}-70a^{20}-67a^{19}-62a^{18}-51a^{17}-36a^{16}-19a^{15}-3a^{14}+16a^{13}+34a^{12}+48a^{11}+57a^{10}+65a^{9}+67a^{8}+61a^{7}+53a^{6}+43a^{5}+29a^{4}+10a^{3}-4a^{2}-20a-206$, $5a^{29}+4a^{28}+5a^{27}+5a^{26}+4a^{25}+4a^{24}+4a^{23}+4a^{22}+4a^{21}+3a^{20}+3a^{19}+5a^{18}+3a^{17}+a^{16}+5a^{15}+4a^{14}+2a^{12}+6a^{11}-a^{9}+6a^{8}+2a^{7}-3a^{6}+4a^{5}+4a^{4}-3a^{3}+5a-17$, $a^{29}-2a^{27}-a^{25}-a^{24}+3a^{23}+a^{22}+3a^{20}-a^{19}-2a^{18}+2a^{17}-a^{16}-a^{15}+a^{14}-3a^{13}+a^{11}-4a^{10}+4a^{9}+5a^{8}-4a^{7}+3a^{6}+2a^{5}-7a^{4}+2a^{3}+a^{2}-4a+2$, $2a^{29}+3a^{28}-4a^{27}-6a^{26}+3a^{25}+8a^{24}-7a^{22}-a^{21}+5a^{20}-2a^{19}-6a^{18}+4a^{17}+10a^{16}-2a^{15}-12a^{14}-a^{13}+9a^{12}+a^{11}-8a^{10}+3a^{9}+12a^{8}-4a^{7}-18a^{6}-a^{5}+19a^{4}+7a^{3}-16a^{2}-5a+8$, $6a^{29}+2a^{28}-2a^{27}-8a^{25}-4a^{23}+2a^{22}+5a^{21}+2a^{20}+7a^{19}-2a^{18}-2a^{17}-4a^{16}-9a^{15}+2a^{14}-3a^{13}+7a^{12}+8a^{11}-a^{10}+11a^{9}-15a^{8}+4a^{7}-14a^{6}-4a^{5}+9a^{4}-8a^{3}+28a^{2}-13a-2$, $2a^{29}+8a^{27}-2a^{26}+2a^{25}+5a^{24}-3a^{23}+3a^{22}+a^{21}-3a^{20}-2a^{19}+3a^{18}-6a^{17}-a^{16}+5a^{15}-13a^{14}+7a^{13}+a^{12}-10a^{11}+13a^{10}-6a^{9}-2a^{8}+9a^{7}-a^{6}-a^{5}+8a^{4}+2a^{3}-12a^{2}+21a-16$
| 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: | \( 56438497078123610 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 56438497078123610 \cdot 2}{2\cdot\sqrt{13601722282800618439068316459204349535511987442947558142619}}\cr\approx \mathstrut & 0.454439961092218 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 3*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^30 - 3*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^30 - 3*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^30 - 3*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 265252859812191058636308480000000 |
| The 5604 conjugacy class representatives for $S_{30}$ are not computed |
| Character table for $S_{30}$ |
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 | $30$ | R | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | $18{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $19{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $25{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{5}$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $30$ | $30$ | $1$ | $30$ | |||
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 13.18.0.1 | $x^{18} + 10 x^{11} + 4 x^{10} + 11 x^{9} + 11 x^{8} + 9 x^{7} + 5 x^{6} + 3 x^{5} + 5 x^{4} + 6 x^{3} + 9 x + 2$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(508\!\cdots\!887\)
| $\Q_{50\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |