Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x + 1)
gp: K = bnfinit(y^30 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x + 1)
\( x^{30} - 4x + 1 \)
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: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2960340583060790267192391089174692605134571100381102676639744\)
\(\medspace = 2^{30}\cdot 7\cdot 149\cdot 8329\cdot 14869697\cdot 24084784266299\cdot 886175540693210057361991\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(103.68\) | 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: |
\(2\), \(7\), \(149\), \(8329\), \(14869697\), \(24084784266299\), \(886175540693210057361991\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{27570\!\cdots\!16831}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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
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: | $15$ | 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: |
$a$, $17a^{29}+7a^{28}-13a^{27}-13a^{26}+4a^{25}+10a^{24}+2a^{23}-a^{22}+2a^{21}-6a^{20}-13a^{19}+2a^{18}+24a^{17}+11a^{16}-24a^{15}-27a^{14}+13a^{13}+32a^{12}+5a^{11}-24a^{10}-12a^{9}+5a^{8}+3a^{7}+3a^{6}+21a^{5}+9a^{4}-34a^{3}-41a^{2}+23a-4$, $6a^{29}-6a^{28}+10a^{27}-9a^{26}+13a^{25}-13a^{24}+15a^{23}-17a^{22}+16a^{21}-20a^{20}+18a^{19}-23a^{18}+22a^{17}-24a^{16}+25a^{15}-22a^{14}+25a^{13}-19a^{12}+22a^{11}-17a^{10}+17a^{9}-16a^{8}+11a^{7}-12a^{6}+a^{5}-a^{4}-10a^{3}+13a^{2}-19a+5$, $10a^{29}+16a^{28}+4a^{27}-9a^{26}-14a^{25}-30a^{24}-28a^{23}-12a^{22}-4a^{21}+19a^{20}+38a^{19}+33a^{18}+32a^{17}+16a^{16}-15a^{15}-26a^{14}-33a^{13}-39a^{12}-15a^{11}+5a^{10}+9a^{9}+26a^{8}+19a^{7}-8a^{6}-2a^{5}-12a^{4}-18a^{3}+27a^{2}+32a-8$, $36a^{29}-11a^{28}+a^{27}-34a^{26}-35a^{25}-4a^{24}-23a^{23}+41a^{22}+16a^{21}+42a^{20}+31a^{19}-25a^{18}+a^{17}-77a^{16}-18a^{15}-22a^{14}-5a^{13}+71a^{12}+11a^{11}+89a^{10}-8a^{9}-19a^{8}-32a^{7}-118a^{6}+14a^{5}-63a^{4}+65a^{3}+72a^{2}+40a-17$, $9a^{29}+38a^{28}+51a^{27}+57a^{26}+51a^{25}+29a^{24}+16a^{23}-5a^{22}-38a^{21}-59a^{20}-81a^{19}-84a^{18}-53a^{17}-23a^{16}+11a^{15}+46a^{14}+64a^{13}+86a^{12}+103a^{11}+88a^{10}+49a^{9}-5a^{8}-67a^{7}-106a^{6}-112a^{5}-119a^{4}-111a^{3}-72a^{2}-32a+14$, $19a^{29}+45a^{28}-3a^{27}-66a^{26}-39a^{25}+50a^{24}+68a^{23}-10a^{22}-59a^{21}-11a^{20}+40a^{19}+7a^{18}-45a^{17}-18a^{16}+52a^{15}+57a^{14}-14a^{13}-70a^{12}-51a^{11}+15a^{10}+67a^{9}+59a^{8}-13a^{7}-77a^{6}-42a^{5}+52a^{4}+50a^{3}-59a^{2}-68a+19$, $27a^{29}-7a^{28}-19a^{27}+7a^{26}+16a^{25}-2a^{24}-25a^{23}+8a^{22}+35a^{21}-26a^{20}-34a^{19}+47a^{18}+24a^{17}-69a^{16}-a^{15}+90a^{14}-45a^{13}-86a^{12}+102a^{11}+40a^{10}-131a^{9}+26a^{8}+119a^{7}-77a^{6}-86a^{5}+111a^{4}+43a^{3}-129a^{2}+14a+5$, $48a^{29}+2a^{28}-55a^{27}+63a^{26}-14a^{25}-53a^{24}+78a^{23}-34a^{22}-46a^{21}+93a^{20}-60a^{19}-32a^{18}+106a^{17}-92a^{16}-8a^{15}+112a^{14}-125a^{13}+25a^{12}+108a^{11}-155a^{10}+64a^{9}+95a^{8}-184a^{7}+113a^{6}+70a^{5}-211a^{4}+176a^{3}+25a^{2}-226a+52$, $4a^{29}+8a^{28}+5a^{27}-2a^{26}-8a^{25}-4a^{24}+a^{23}+4a^{22}+3a^{21}-2a^{19}+a^{18}+2a^{17}-3a^{16}-5a^{15}-6a^{14}+4a^{13}+13a^{12}+10a^{11}-5a^{10}-17a^{9}-17a^{8}+a^{7}+24a^{6}+21a^{5}+4a^{4}-22a^{3}-29a^{2}-8a+6$, $14a^{29}-18a^{28}+27a^{27}-a^{26}-12a^{25}+9a^{24}-37a^{23}+41a^{22}-13a^{21}+17a^{20}-3a^{19}-44a^{18}+38a^{17}-30a^{16}+54a^{15}-16a^{14}-28a^{13}+17a^{12}-46a^{11}+74a^{10}-22a^{9}+3a^{8}-3a^{7}-67a^{6}+78a^{5}-35a^{4}+50a^{3}-9a^{2}-82a+21$, $a^{29}-4a^{28}+9a^{27}+2a^{26}+5a^{25}-6a^{24}-3a^{23}+a^{22}+5a^{21}+10a^{20}-2a^{19}+a^{18}-7a^{17}+9a^{16}+8a^{15}+11a^{14}-a^{13}-8a^{12}-2a^{11}+4a^{10}+14a^{9}+5a^{8}-5a^{7}-6a^{6}-6a^{5}+22a^{4}+7a^{3}+24a^{2}-23a+6$, $14a^{29}+12a^{28}-9a^{27}-16a^{26}-3a^{25}+25a^{24}+10a^{23}-39a^{22}-4a^{21}+52a^{20}+4a^{19}-53a^{18}+13a^{17}+54a^{16}-8a^{15}-32a^{14}+3a^{13}+33a^{12}+23a^{11}-25a^{10}-48a^{9}+45a^{8}+67a^{7}-71a^{6}-72a^{5}+89a^{4}+57a^{3}-102a^{2}-40a+16$, $3a^{29}+a^{28}+4a^{27}+18a^{26}+15a^{25}+23a^{24}+3a^{23}+5a^{22}-3a^{21}+8a^{20}+13a^{19}+9a^{18}+3a^{17}-17a^{16}-11a^{15}-25a^{14}-3a^{13}-21a^{12}-8a^{11}-22a^{10}-21a^{9}-13a^{8}-29a^{7}-8a^{6}-26a^{5}+13a^{4}+3a^{3}+33a^{2}+4a-4$, $5a^{29}+3a^{28}-3a^{26}-4a^{25}-3a^{24}-a^{23}-a^{22}-a^{21}+2a^{20}+4a^{19}+2a^{18}-a^{17}-3a^{16}-2a^{15}+a^{14}+4a^{13}+2a^{12}+3a^{11}+5a^{10}+3a^{9}-5a^{8}-11a^{7}-12a^{6}-8a^{5}+2a^{4}+5a^{3}+7a^{2}+10a-4$
| 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: | \( 7912688478084103000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 7912688478084103000 \cdot 1}{2\cdot\sqrt{2960340583060790267192391089174692605134571100381102676639744}}\cr\approx \mathstrut & 1.37467926221266 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 4*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^30 - 4*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^30 - 4*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^30 - 4*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 265252859812191058636308480000000 |
| The 5604 conjugacy class representatives for $S_{30}$ are not computed |
| Character table for $S_{30}$ 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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/5.5.0.1}{5} }$ | R | $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $15^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $22{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.11.0.1 | $x^{11} + x + 4$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
|
\(149\)
| $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 149.2.1.2 | $x^{2} + 298$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 149.5.0.1 | $x^{5} + 2 x + 147$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 149.7.0.1 | $x^{7} + 19 x + 147$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 149.14.0.1 | $x^{14} - x + 43$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(8329\)
| $\Q_{8329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{8329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(14869697\)
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(24084784266299\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
|
\(886\!\cdots\!991\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |