Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 2*x - 4)
gp: K = bnfinit(y^31 - 2*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 2*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 2*x - 4)
\( x^{31} - 2x - 4 \)
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: |
\(-4919854480186888668438405507521989689337153398951757258543857664\)
\(\medspace = -\,2^{58}\cdot 18637\cdot 91\!\cdots\!13\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(113.39\) | 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: |
\(2\), \(18637\), \(91587\!\cdots\!45713\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-17069\!\cdots\!53181}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{2}a^{30}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: |
$a^{29}-3a^{28}+a^{27}-a^{26}-a^{25}+2a^{24}+a^{22}+a^{21}-2a^{19}-3a^{17}-2a^{16}+2a^{15}-3a^{14}+4a^{13}+2a^{12}-a^{11}+4a^{10}-3a^{9}-2a^{8}-a^{7}-3a^{6}-a^{5}+3a^{4}+2a^{3}+3a^{2}+7a-3$, $2a^{30}-6a^{29}-5a^{28}+3a^{27}-3a^{26}-9a^{25}+a^{24}+a^{23}-10a^{22}-4a^{21}+5a^{20}-8a^{19}-10a^{18}+4a^{17}-3a^{16}-14a^{15}-2a^{14}+2a^{13}-14a^{12}-11a^{11}+3a^{10}-8a^{9}-18a^{8}-2a^{7}+a^{6}-20a^{5}-13a^{4}+6a^{3}-14a^{2}-25a-3$, $7a^{30}+6a^{28}+5a^{27}-6a^{26}-a^{25}-a^{24}-12a^{23}-6a^{22}-2a^{21}-12a^{20}-3a^{19}+5a^{18}-5a^{17}+4a^{16}+13a^{15}-2a^{14}+2a^{13}+11a^{12}-9a^{11}-9a^{10}+2a^{9}-17a^{8}-17a^{7}+4a^{6}-9a^{5}-4a^{4}+25a^{3}+15a^{2}+14a+31$, $5a^{30}+4a^{29}+4a^{28}+a^{27}-a^{26}-6a^{25}-6a^{24}-5a^{23}-4a^{22}+5a^{21}+4a^{20}+10a^{19}+7a^{18}+3a^{17}-a^{16}-10a^{15}-9a^{14}-12a^{13}-4a^{12}+a^{11}+9a^{10}+17a^{9}+12a^{8}+11a^{7}-3a^{6}-14a^{5}-16a^{4}-23a^{3}-10a^{2}+a+3$, $4a^{29}-7a^{28}+a^{27}+7a^{26}-7a^{25}+6a^{23}-3a^{22}-a^{21}+3a^{20}-2a^{19}-5a^{18}+7a^{17}-11a^{15}+11a^{14}+3a^{13}-9a^{12}+8a^{11}-3a^{10}-4a^{9}+7a^{8}-6a^{7}-6a^{6}+7a^{5}+5a^{4}-7a^{3}+3a^{2}+7a-11$, $19a^{30}-21a^{29}-6a^{28}+28a^{27}-20a^{26}-7a^{25}+12a^{24}+a^{23}-5a^{22}-14a^{21}+24a^{20}-9a^{19}-23a^{18}+34a^{17}-18a^{16}-12a^{15}+23a^{14}-16a^{13}-2a^{12}-3a^{11}+15a^{10}-7a^{9}-37a^{8}+51a^{7}-10a^{6}-52a^{5}+45a^{4}+8a^{3}-27a^{2}-25a-5$, $27a^{30}-37a^{28}-49a^{27}-32a^{26}+7a^{25}+49a^{24}+57a^{23}+33a^{22}-21a^{21}-64a^{20}-68a^{19}-28a^{18}+43a^{17}+90a^{16}+91a^{15}+32a^{14}-51a^{13}-99a^{12}-87a^{11}+a^{10}+98a^{9}+153a^{8}+118a^{7}+2a^{6}-113a^{5}-172a^{4}-103a^{3}+36a^{2}+177a+163$, $5a^{29}+11a^{28}-14a^{27}-5a^{26}+23a^{25}-3a^{24}-10a^{23}-7a^{22}+27a^{21}+3a^{20}-26a^{19}+11a^{18}+15a^{17}+5a^{16}-27a^{15}+14a^{14}+29a^{13}-23a^{12}-10a^{11}+17a^{10}+37a^{9}-31a^{8}-26a^{7}+48a^{6}+18a^{5}-22a^{4}-32a^{3}+54a^{2}+22a-49$, $16a^{30}-5a^{29}-7a^{28}+14a^{27}-12a^{25}+16a^{24}+7a^{23}-17a^{22}+21a^{21}+10a^{20}-21a^{19}+24a^{18}+9a^{17}-22a^{16}+23a^{15}+11a^{14}-20a^{13}+18a^{12}+19a^{11}-21a^{10}+12a^{9}+31a^{8}-28a^{7}+11a^{6}+45a^{5}-34a^{4}+16a^{3}+55a^{2}-35a-17$, $6a^{30}+5a^{29}-a^{28}-11a^{27}-18a^{26}-14a^{25}+a^{24}+18a^{23}+22a^{22}+6a^{21}-20a^{20}-36a^{19}-25a^{18}+9a^{17}+42a^{16}+49a^{15}+22a^{14}-19a^{13}-43a^{12}-33a^{11}+3a^{10}+36a^{9}+43a^{8}+23a^{7}-7a^{6}-25a^{5}-25a^{4}-14a^{3}+13a+15$, $14a^{30}-13a^{29}+4a^{28}+14a^{27}-13a^{26}+3a^{25}+11a^{24}-9a^{23}-6a^{22}+11a^{21}-7a^{20}-18a^{19}+18a^{18}-11a^{17}-23a^{16}+26a^{15}-17a^{14}-21a^{13}+26a^{12}-16a^{11}-19a^{10}+22a^{9}-2a^{8}-21a^{7}+25a^{6}+16a^{5}-26a^{4}+33a^{3}+26a^{2}-35a+11$, $8a^{30}+4a^{29}-13a^{28}+10a^{27}-21a^{26}+23a^{25}+5a^{24}-10a^{23}-5a^{22}-17a^{21}+35a^{20}-9a^{19}-2a^{18}-12a^{17}-a^{16}+25a^{15}-19a^{14}+21a^{13}-32a^{12}-4a^{11}+19a^{10}+9a^{9}+30a^{8}-63a^{7}+3a^{6}+19a^{5}+24a^{4}-2a^{3}-54a^{2}+31a-23$, $31a^{30}-26a^{29}-25a^{28}+40a^{27}-12a^{26}-27a^{25}+38a^{24}-20a^{23}-41a^{22}+51a^{21}+3a^{20}-44a^{19}+41a^{18}-11a^{17}-65a^{16}+57a^{15}+30a^{14}-58a^{13}+41a^{12}+4a^{11}-98a^{10}+47a^{9}+61a^{8}-72a^{7}+36a^{6}+31a^{5}-133a^{4}+18a^{3}+93a^{2}-90a-47$, $10a^{30}+11a^{29}+3a^{28}+25a^{27}-4a^{26}+38a^{25}-14a^{24}+46a^{23}-30a^{22}+48a^{21}-50a^{20}+47a^{19}-67a^{18}+48a^{17}-79a^{16}+50a^{15}-89a^{14}+47a^{13}-99a^{12}+40a^{11}-105a^{10}+36a^{9}-99a^{8}+41a^{7}-74a^{6}+56a^{5}-39a^{4}+70a^{3}-7a^{2}+68a-1$, $18a^{30}+58a^{29}+61a^{28}+67a^{27}+73a^{26}+52a^{25}+58a^{24}+16a^{23}+16a^{22}-43a^{21}-27a^{20}-58a^{19}-42a^{18}-42a^{17}-31a^{16}-4a^{15}+7a^{14}+54a^{13}+13a^{12}+37a^{11}-27a^{10}-28a^{9}-94a^{8}-126a^{7}-191a^{6}-213a^{5}-179a^{4}-176a^{3}-70a^{2}-32a+75$
| 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: | \( 210828920582287920000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 210828920582287920000 \cdot 1}{2\cdot\sqrt{4919854480186888668438405507521989689337153398951757258543857664}}\cr\approx \mathstrut & 2.82261783877076 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 2*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 - 2*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 - 2*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 - 2*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 $S_{31}$ |
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 | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/19.9.0.1}{9} }$ | $30{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $18{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| Deg $30$ | $30$ | $1$ | $58$ | ||||
|
\(18637\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(915\!\cdots\!713\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ |