Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x + 4)
gp: K = bnfinit(y^30 - 3*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 3*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 3*x + 4)
\( x^{30} - 3x + 4 \)
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: |
\(-59343549786133513466171876363248204349535511987442947558142619\)
\(\medspace = -\,3^{30}\cdot 4329008673689392661\cdot 66580556933815696116855581671\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(114.58\) | 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: |
\(3\), \(4329008673689392661\), \(66580556933815696116855581671\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-28822\!\cdots\!16531}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+2a^{9}-2a^{8}+2a^{7}-2a^{6}+2a^{5}-2a^{4}+2a^{3}-2a^{2}+2a-1$, $a^{27}-a^{26}+2a^{25}-2a^{24}+2a^{23}-2a^{22}+a^{21}-a^{20}+a^{15}-a^{14}+2a^{13}-2a^{12}+2a^{11}-2a^{10}+a^{9}-a^{8}+a^{3}-a^{2}+2a-1$, $2a^{29}-a^{28}+a^{26}-2a^{25}+a^{24}-2a^{23}+a^{20}-a^{19}+3a^{18}-2a^{17}+2a^{16}+a^{15}-4a^{14}+5a^{13}-6a^{12}+4a^{11}-3a^{10}-a^{9}+a^{8}+a^{6}+2a^{5}-2a^{3}+7a^{2}-11a+5$, $a^{21}-2a^{20}-2a^{11}+3a^{10}-a^{2}+5a-5$, $11a^{29}-3a^{28}+6a^{27}-20a^{26}+9a^{25}-5a^{24}+20a^{23}-5a^{22}-8a^{21}-8a^{20}-8a^{19}+26a^{18}-8a^{17}+19a^{16}-38a^{15}+15a^{14}-10a^{13}+27a^{12}+2a^{11}-21a^{10}+a^{9}-30a^{8}+56a^{7}-24a^{6}+41a^{5}-63a^{4}+15a^{3}-15a^{2}+30a-1$, $3a^{29}+2a^{28}+a^{27}-3a^{26}+a^{25}-4a^{24}-a^{23}-9a^{22}-5a^{21}-7a^{20}-4a^{19}-7a^{18}-13a^{17}-7a^{16}-11a^{15}-a^{14}-14a^{13}-8a^{12}-15a^{11}-4a^{10}-7a^{9}-12a^{8}-8a^{7}-17a^{6}+2a^{5}-17a^{4}-a^{3}-19a^{2}+a-19$, $15a^{29}-17a^{28}-17a^{27}+12a^{26}+19a^{25}-7a^{24}-20a^{23}+8a^{22}+8a^{21}+10a^{20}-5a^{19}-6a^{18}-20a^{17}+18a^{16}+28a^{15}-21a^{14}-38a^{13}+6a^{12}+61a^{11}-23a^{10}-31a^{9}-9a^{8}+41a^{7}+5a^{6}-10a^{5}-2a^{4}-44a^{3}+34a^{2}+37a-35$, $22a^{29}+15a^{28}+3a^{27}+a^{26}-16a^{25}-26a^{24}-13a^{23}-a^{22}+14a^{21}+27a^{20}+14a^{19}+11a^{18}+2a^{17}-32a^{16}-32a^{15}-8a^{14}-5a^{13}+21a^{12}+32a^{11}+19a^{10}+25a^{9}-11a^{8}-54a^{7}-21a^{6}-15a^{5}-15a^{4}+36a^{3}+39a^{2}+36a-35$, $12a^{29}-38a^{28}-32a^{27}+49a^{26}+41a^{25}-70a^{24}-29a^{23}+62a^{22}+35a^{21}-49a^{20}-68a^{19}+75a^{18}+76a^{17}-100a^{16}-55a^{15}+76a^{14}+74a^{13}-64a^{12}-129a^{11}+110a^{10}+125a^{9}-132a^{8}-92a^{7}+81a^{6}+146a^{5}-73a^{4}-224a^{3}+150a^{2}+193a-189$, $14a^{29}+4a^{28}-12a^{27}+9a^{26}-4a^{25}+10a^{24}+17a^{23}-21a^{22}+8a^{21}-5a^{20}+27a^{18}-19a^{17}+a^{16}-21a^{14}+35a^{13}-5a^{12}-13a^{11}+10a^{10}-44a^{9}+33a^{8}+17a^{7}-29a^{6}+23a^{5}-48a^{4}-3a^{3}+54a^{2}-54a+11$, $3a^{29}-a^{28}+a^{27}-5a^{26}-8a^{25}-2a^{24}-4a^{23}-a^{22}+a^{21}+2a^{20}+11a^{19}+4a^{18}+2a^{17}+6a^{16}-3a^{15}-a^{14}-9a^{13}-11a^{12}+2a^{11}-8a^{10}-4a^{9}+8a^{8}+3a^{7}+14a^{6}+a^{5}+2a^{4}+14a^{3}-11a^{2}-8a-15$, $9a^{29}-9a^{28}+a^{27}+7a^{26}-9a^{25}+3a^{24}+a^{23}-4a^{22}+6a^{21}-5a^{20}+2a^{19}+5a^{18}-8a^{17}+5a^{16}+a^{15}-4a^{14}+4a^{13}-4a^{12}+a^{11}+a^{10}-3a^{9}+3a^{7}+7a^{6}-17a^{5}+10a^{4}+20a^{3}-44a^{2}+25a+1$, $19a^{29}+11a^{28}-14a^{26}-22a^{25}-18a^{24}-9a^{23}+7a^{22}+20a^{21}+23a^{20}+15a^{19}-2a^{18}-18a^{17}-25a^{16}-24a^{15}-6a^{14}+13a^{13}+29a^{12}+35a^{11}+15a^{10}-24a^{8}-39a^{7}-28a^{6}-12a^{5}+17a^{4}+41a^{3}+34a^{2}+22a-59$, $10a^{29}-20a^{28}+6a^{27}+12a^{26}-23a^{25}+8a^{24}+30a^{23}-23a^{22}-11a^{21}+32a^{20}-17a^{19}-16a^{18}+46a^{17}-12a^{16}-48a^{15}+40a^{14}+15a^{13}-54a^{12}+34a^{11}+29a^{10}-77a^{9}+13a^{8}+74a^{7}-64a^{6}-20a^{5}+79a^{4}-57a^{3}-45a^{2}+119a-55$
| 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: | \( 9555863009872810000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 9555863009872810000 \cdot 2}{2\cdot\sqrt{59343549786133513466171876363248204349535511987442947558142619}}\cr\approx \mathstrut & 1.16487997616482 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x + 4)
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 + 4, 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 + 4);
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 + 4);
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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | $25{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $21{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $30$ | $29{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
| 3.12.12.18 | $x^{12} - 36 x^{11} + 438 x^{10} + 120 x^{9} - 4563 x^{8} + 1188 x^{7} + 22410 x^{6} + 10692 x^{5} - 17658 x^{4} - 3780 x^{3} + 6804 x^{2} - 1296 x + 81$ | $3$ | $4$ | $12$ | 12T46 | $[3/2, 3/2]_{2}^{4}$ | |
| 3.12.12.20 | $x^{12} + 30 x^{10} + 228 x^{9} + 1872 x^{8} + 5778 x^{7} + 15336 x^{6} + 18036 x^{5} + 12879 x^{4} + 7074 x^{3} + 3240 x^{2} + 810 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
|
\(4329008673689392661\)
| $\Q_{4329008673689392661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
|
\(665\!\cdots\!671\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |