Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + x - 4)
gp: K = bnfinit(y^31 + y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + x - 4)
\( x^{31} + x - 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-19679417921189702513400979877982159757348613595807029034175430656\)
\(\medspace = -\,2^{30}\cdot 103\cdot 1946342156463328037\cdot 91\!\cdots\!79\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.58\) | 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: |
\(2\), \(103\), \(1946342156463328037\), \(91423\!\cdots\!48779\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-18327\!\cdots\!32769}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$
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: |
$a^{30}-a^{29}-2a^{28}+a^{27}+3a^{26}-a^{25}-3a^{24}-3a^{23}+2a^{22}+7a^{21}+2a^{20}-7a^{19}-7a^{18}-a^{17}+10a^{16}+9a^{15}-4a^{14}-12a^{13}-9a^{12}+9a^{11}+13a^{10}+4a^{9}-9a^{8}-15a^{7}+14a^{5}+7a^{4}-3a^{3}-10a^{2}-7a+11$, $2a^{30}-4a^{29}-4a^{28}-a^{27}+2a^{26}+4a^{25}+3a^{24}-4a^{23}-4a^{22}-5a^{21}+4a^{20}+5a^{19}+3a^{18}+2a^{17}-10a^{16}-4a^{14}+13a^{13}+a^{12}+3a^{11}-8a^{10}-5a^{9}-3a^{8}+12a^{7}+4a^{6}+9a^{5}-19a^{4}+3a^{3}-17a^{2}+18a+5$, $18a^{30}-31a^{29}+13a^{28}-9a^{27}+41a^{26}-51a^{25}+14a^{24}+2a^{23}+25a^{22}-23a^{21}-24a^{20}+48a^{19}-28a^{18}+19a^{17}-53a^{16}+82a^{15}-50a^{14}+a^{13}-18a^{12}+50a^{11}+8a^{10}-86a^{9}+63a^{8}-20a^{7}+77a^{6}-141a^{5}+95a^{4}-20a^{3}+27a^{2}-67a+37$, $23a^{30}-20a^{29}-23a^{28}+28a^{27}+18a^{26}-27a^{25}-19a^{24}+44a^{23}+12a^{22}-43a^{21}-24a^{20}+57a^{19}+9a^{18}-53a^{17}-24a^{16}+68a^{15}+5a^{14}-56a^{13}-12a^{12}+74a^{11}-17a^{10}-72a^{9}+10a^{8}+92a^{7}-34a^{6}-92a^{5}+38a^{4}+117a^{3}-40a^{2}-113a+77$, $24a^{30}-3a^{29}-29a^{28}-29a^{27}-25a^{26}+8a^{25}+35a^{24}+39a^{23}+23a^{22}-8a^{21}-42a^{20}-48a^{19}-17a^{18}+8a^{17}+54a^{16}+54a^{15}+15a^{14}-15a^{13}-64a^{12}-63a^{11}-16a^{10}+26a^{9}+69a^{8}+72a^{7}+15a^{6}-44a^{5}-76a^{4}-83a^{3}-18a^{2}+68a+105$, $14a^{30}+10a^{29}-22a^{28}+10a^{27}+8a^{26}-17a^{25}+20a^{24}-2a^{23}-29a^{22}+25a^{21}+8a^{20}-16a^{19}+17a^{18}-18a^{17}-20a^{16}+29a^{15}-6a^{13}+6a^{12}-16a^{11}+14a^{10}+10a^{9}-13a^{8}+7a^{7}-24a^{6}+7a^{5}+51a^{4}-39a^{3}-12a^{2}+24a-51$, $5a^{30}+15a^{29}+9a^{28}+9a^{27}+4a^{26}+a^{25}+4a^{24}-14a^{23}-6a^{22}-25a^{21}-22a^{19}+3a^{18}-15a^{17}+16a^{16}+8a^{15}+17a^{14}+17a^{13}+11a^{12}+31a^{11}-4a^{10}+13a^{9}-30a^{8}+4a^{7}-32a^{6}-13a^{5}-38a^{4}-15a^{3}-3a^{2}-4a+23$, $10a^{30}+6a^{29}-2a^{28}-2a^{27}-12a^{26}-2a^{24}+16a^{23}+7a^{22}+7a^{21}-9a^{20}-15a^{19}-13a^{18}-9a^{17}+16a^{16}+13a^{15}+32a^{14}+a^{13}-5a^{12}-30a^{11}-20a^{10}-15a^{9}+4a^{8}+36a^{7}+22a^{6}+26a^{5}-24a^{4}-6a^{3}-40a^{2}-8a+13$, $6a^{30}+3a^{29}-3a^{28}+9a^{27}-6a^{26}-10a^{25}-7a^{24}-2a^{23}+a^{22}-19a^{21}+3a^{19}+2a^{18}+a^{17}+2a^{16}+26a^{15}-3a^{14}-2a^{13}+3a^{12}+5a^{11}+a^{10}-30a^{9}+5a^{8}-10a^{7}-20a^{6}-17a^{5}-3a^{4}+34a^{3}-20a^{2}+3a+23$, $4a^{30}+4a^{29}+10a^{28}+10a^{27}+9a^{26}+8a^{25}+6a^{24}+6a^{23}+8a^{22}+14a^{21}+12a^{20}+8a^{19}+7a^{18}+4a^{17}+13a^{16}+14a^{15}+12a^{14}+20a^{13}+6a^{12}+6a^{11}+18a^{10}+12a^{9}+26a^{8}+19a^{7}+3a^{6}+19a^{5}+8a^{4}+13a^{3}+33a^{2}+16a+21$, $11a^{30}-7a^{29}+30a^{28}+20a^{27}-12a^{26}+28a^{25}+27a^{24}-20a^{23}+18a^{22}+33a^{21}-24a^{20}+5a^{19}+31a^{18}-33a^{17}-13a^{16}+22a^{15}-50a^{14}-34a^{13}+24a^{12}-56a^{11}-52a^{10}+39a^{9}-41a^{8}-67a^{7}+47a^{6}-17a^{5}-74a^{4}+52a^{3}+12a^{2}-65a+87$, $a^{29}-a^{27}+a^{25}-a^{23}+a^{21}-a^{19}+a^{17}-a^{15}+a^{13}-a^{11}+a^{9}-a^{7}+a^{5}-a^{3}-a^{2}+5a-5$, $2a^{30}-6a^{29}-a^{28}-8a^{27}-6a^{25}+a^{24}-7a^{23}-3a^{22}-8a^{21}-4a^{19}-5a^{18}-7a^{17}-7a^{16}-2a^{15}-7a^{14}-14a^{12}-2a^{11}-13a^{10}+8a^{9}-17a^{8}+2a^{7}-22a^{6}+7a^{5}-19a^{4}+10a^{3}-25a^{2}+a-25$, $37a^{30}+29a^{29}-21a^{28}-50a^{27}+29a^{26}+40a^{25}-21a^{24}-8a^{23}-16a^{22}-14a^{21}+22a^{20}-14a^{19}+17a^{18}+57a^{17}-52a^{16}-53a^{15}+42a^{14}+44a^{13}+31a^{12}-85a^{11}-133a^{10}+107a^{9}+162a^{8}-40a^{7}-113a^{6}-91a^{5}+103a^{4}+216a^{3}-114a^{2}-235a+63$, $8a^{30}+9a^{29}+a^{28}-11a^{27}-8a^{26}+4a^{25}+2a^{24}-a^{22}+16a^{21}+a^{20}-7a^{19}-23a^{18}+7a^{17}+3a^{16}+15a^{15}-11a^{14}+15a^{13}-4a^{12}+8a^{11}-31a^{10}-a^{9}-7a^{8}+36a^{7}+2a^{6}+8a^{5}-28a^{4}+8a^{3}-9a^{2}-15$
| 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: | \( 367578496614054170000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 367578496614054170000 \cdot 1}{2\cdot\sqrt{19679417921189702513400979877982159757348613595807029034175430656}}\cr\approx \mathstrut & 2.46060554409269 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 + 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^31 + 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^31 + 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^31 + 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 8222838654177922817725562880000000 |
| The 6842 conjugacy class representatives for $S_{31}$ are not computed |
| Character table for $S_{31}$ 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 | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $31$ | $25{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $17{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $31$ | ${\href{/padicField/31.2.0.1}{2} }^{15}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/47.7.0.1}{7} }$ | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(103\)
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 103.5.0.1 | $x^{5} + 11 x + 98$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 103.6.0.1 | $x^{6} + 96 x^{3} + 9 x^{2} + 30 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 103.18.0.1 | $x^{18} - 19 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(1946342156463328037\)
| $\Q_{1946342156463328037}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1946342156463328037}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1946342156463328037}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(914\!\cdots\!779\)
| $\Q_{91\!\cdots\!79}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ |