Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x - 3)
gp: K = bnfinit(y^30 - 4*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 4*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x - 3)
\( x^{30} - 4x - 3 \)
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: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2974470969152529207588287994890692605134571100381102676639744\)
\(\medspace = 2^{30}\cdot 47\cdot 113\cdot 70468441\cdot 118739592888857\cdot 62336633454993137535840733\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(103.70\) | 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: |
\(2\), \(47\), \(113\), \(70468441\), \(118739592888857\), \(62336633454993137535840733\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{27701\!\cdots\!29331}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{29}-a^{27}+a^{25}-a^{23}+a^{21}-a^{19}+a^{17}-2a^{15}-2a^{14}+a^{13}+2a^{12}-a^{11}-2a^{10}+a^{9}+2a^{8}-a^{7}-2a^{6}+a^{5}+2a^{4}-a^{3}-2a^{2}+2a+2$, $a^{29}-2a^{28}+a^{26}+a^{25}-2a^{24}+a^{22}+a^{21}-2a^{20}+a^{18}+a^{17}-2a^{16}+a^{14}+3a^{13}-4a^{11}+a^{10}+3a^{9}-4a^{7}+a^{6}+3a^{5}-4a^{3}+a^{2}+3a-4$, $9a^{29}+6a^{28}-3a^{27}-12a^{26}-12a^{25}-4a^{24}+6a^{23}+14a^{22}+13a^{21}-14a^{19}-18a^{18}-11a^{17}+5a^{16}+22a^{15}+24a^{14}+7a^{13}-15a^{12}-29a^{11}-27a^{10}-3a^{9}+28a^{8}+39a^{7}+24a^{6}-6a^{5}-38a^{4}-46a^{3}-14a^{2}+32a+22$, $4a^{29}+4a^{28}-6a^{26}-2a^{25}+4a^{24}+5a^{23}-a^{22}-6a^{21}-a^{20}+4a^{19}+5a^{18}-4a^{17}-6a^{16}+a^{15}+7a^{14}+4a^{13}-10a^{12}-7a^{11}+3a^{10}+15a^{9}+2a^{8}-14a^{7}-15a^{6}+8a^{5}+23a^{4}+5a^{3}-19a^{2}-23a-5$, $10a^{29}-11a^{28}+4a^{27}-6a^{26}+3a^{25}-3a^{24}+3a^{23}-3a^{22}-5a^{20}-3a^{19}-4a^{18}-2a^{17}+a^{16}+3a^{15}+6a^{14}+6a^{13}+5a^{12}+4a^{11}+2a^{7}+7a^{6}+9a^{5}+11a^{4}+8a^{3}+3a^{2}-3a-52$, $18a^{29}-19a^{28}+18a^{27}-15a^{26}+13a^{25}-12a^{24}+13a^{23}-15a^{22}+16a^{21}-14a^{20}+11a^{19}-5a^{18}+a^{17}+a^{16}-a^{15}-2a^{14}+3a^{13}-a^{12}-4a^{11}+11a^{10}-18a^{9}+18a^{8}-18a^{7}+12a^{6}-9a^{5}+10a^{4}-13a^{3}+20a^{2}-26a-46$, $a^{29}+a^{28}-a^{27}-2a^{26}-3a^{25}-2a^{24}+3a^{22}+5a^{21}+4a^{20}+3a^{19}-3a^{18}-4a^{17}-6a^{16}-4a^{15}+2a^{14}+3a^{13}+8a^{12}+5a^{11}+2a^{10}-5a^{8}-3a^{7}-4a^{6}-2a^{5}+2a^{3}+5a^{2}+9a+5$, $6a^{29}-4a^{28}+6a^{27}+a^{26}+6a^{24}-5a^{23}+3a^{22}-2a^{21}-8a^{20}+5a^{19}-16a^{18}+8a^{17}-11a^{16}+a^{15}+5a^{14}-7a^{13}+17a^{12}-8a^{11}+13a^{10}+2a^{9}-3a^{8}+9a^{7}-15a^{6}+5a^{5}-10a^{4}-8a^{3}+5a^{2}-12a-8$, $32a^{29}-37a^{28}+18a^{27}-16a^{26}+31a^{25}-30a^{24}+12a^{23}-6a^{22}+24a^{21}-26a^{20}-8a^{19}+9a^{18}+28a^{17}-7a^{16}-36a^{15}+6a^{14}+20a^{13}+24a^{12}-43a^{11}+8a^{10}+a^{9}+32a^{8}-54a^{7}+19a^{6}+11a^{5}+48a^{4}-76a^{3}-10a^{2}+15a-44$, $3a^{29}+a^{28}-a^{27}+a^{25}-3a^{24}-4a^{23}-3a^{22}-4a^{21}-2a^{20}+2a^{19}-2a^{17}+3a^{16}+4a^{15}-a^{14}-a^{13}-a^{12}-4a^{11}-2a^{10}+a^{9}-7a^{8}-9a^{7}-2a^{6}-3a^{5}-9a^{4}-4a^{3}+a-1$, $5a^{29}-4a^{28}+4a^{27}+2a^{26}-5a^{25}+7a^{24}-a^{23}+2a^{22}+6a^{20}-4a^{19}+7a^{18}-4a^{16}+11a^{15}-3a^{14}+3a^{13}+3a^{12}+8a^{11}-3a^{10}+10a^{9}-a^{8}-a^{7}+15a^{6}-6a^{5}+4a^{4}+10a^{3}+8a^{2}-a-5$, $23a^{29}+7a^{28}-36a^{27}+62a^{26}-81a^{25}+92a^{24}-91a^{23}+78a^{22}-57a^{21}+27a^{20}+7a^{19}-42a^{18}+73a^{17}-93a^{16}+105a^{15}-105a^{14}+92a^{13}-66a^{12}+28a^{11}+9a^{10}-46a^{9}+81a^{8}-108a^{7}+125a^{6}-120a^{5}+102a^{4}-75a^{3}+37a^{2}+5a-148$, $32a^{29}+10a^{28}-a^{27}-37a^{26}+11a^{25}+30a^{24}+a^{23}-42a^{22}+a^{21}+44a^{20}+11a^{19}-46a^{18}-20a^{17}+43a^{16}+28a^{15}-50a^{14}-40a^{13}+54a^{12}+56a^{11}-46a^{10}-63a^{9}+38a^{8}+72a^{7}-36a^{6}-96a^{5}+24a^{4}+116a^{3}-8a^{2}-123a-130$, $9a^{29}-29a^{28}+13a^{27}+21a^{26}-37a^{25}+48a^{23}-20a^{22}-28a^{21}+43a^{20}-5a^{19}-48a^{18}+36a^{17}+27a^{16}-44a^{15}+27a^{14}+42a^{13}-68a^{12}-13a^{11}+78a^{10}-35a^{9}-45a^{8}+96a^{7}+10a^{6}-101a^{5}+41a^{4}+51a^{3}-84a^{2}+39a+73$, $136a^{29}-96a^{28}+74a^{27}-50a^{26}+32a^{25}-34a^{24}+10a^{23}-16a^{22}-3a^{21}-11a^{20}-2a^{19}+7a^{18}+4a^{17}+9a^{16}+6a^{15}+19a^{14}+6a^{13}+a^{12}-9a^{11}-5a^{10}-15a^{9}-25a^{8}-27a^{7}-17a^{6}-3a^{5}-10a^{4}+5a^{3}+17a^{2}+41a-523$
| 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: | \( 10204871904021840000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 10204871904021840000 \cdot 1}{2\cdot\sqrt{2974470969152529207588287994890692605134571100381102676639744}}\cr\approx \mathstrut & 1.76868642698741 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x - 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 - 4*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 - 4*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 - 4*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 $S_{30}$ 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 | $28{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $30$ | $27{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | R | $15{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 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 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.5.0.1 | $x^{5} + x + 42$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 47.18.0.1 | $x^{18} + 6 x^{11} + 41 x^{10} + 42 x^{9} + 26 x^{8} + 44 x^{7} + 24 x^{6} + 22 x^{5} + 11 x^{4} + 5 x^{3} + 45 x^{2} + 33 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(113\)
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
|
\(70468441\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
|
\(118739592888857\)
| $\Q_{118739592888857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{118739592888857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{118739592888857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(623\!\cdots\!733\)
| $\Q_{62\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |