Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 3)
gp: K = bnfinit(y^31 - 5*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 5*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 5*x - 3)
\( x^{31} - 5x - 3 \)
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: | $[3, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(958751781772834738039164297352952377530037676210703017156038840281\)
\(\medspace = 3^{30}\cdot 13\cdot 16690100963\cdot 21\!\cdots\!51\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(134.41\) | 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: |
\(3\), \(13\), \(16690100963\), \(21461\!\cdots\!40951\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{46565\!\cdots\!65569}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $16$ | 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: |
$a+1$, $5a^{30}-5a^{29}+2a^{27}-6a^{26}+7a^{25}-8a^{24}+3a^{23}-2a^{22}-2a^{21}+6a^{20}-7a^{19}+6a^{18}-5a^{17}+2a^{16}+a^{15}+a^{14}+4a^{13}-3a^{12}+a^{11}-3a^{10}+6a^{9}-4a^{8}+6a^{7}-5a^{6}-4a^{5}-12a^{3}+14a^{2}-16a-23$, $6a^{30}-8a^{29}+7a^{28}-6a^{27}+2a^{26}-5a^{25}+12a^{24}-13a^{23}+15a^{22}-12a^{21}+5a^{20}-2a^{19}-5a^{18}+4a^{17}+5a^{16}-2a^{15}-2a^{14}+10a^{13}-21a^{12}+18a^{11}-20a^{10}+17a^{9}+3a^{7}-9a^{6}+14a^{5}-24a^{4}+7a^{3}+6a^{2}-9a-1$, $a^{30}+11a^{29}-3a^{28}-7a^{27}-13a^{26}-8a^{25}+2a^{24}+11a^{23}+14a^{22}+6a^{21}-6a^{20}-15a^{19}-13a^{18}+3a^{17}+21a^{16}+33a^{15}+25a^{14}+3a^{13}-24a^{12}-37a^{11}-24a^{10}+6a^{9}+41a^{8}+51a^{7}+31a^{6}-15a^{5}-59a^{4}-73a^{3}-49a^{2}+6a+44$, $a^{30}+a^{29}-4a^{28}-4a^{27}-6a^{26}-4a^{25}+a^{24}+a^{23}+7a^{22}+6a^{21}+5a^{20}-4a^{18}-6a^{17}-13a^{16}-6a^{15}-5a^{14}+4a^{13}+9a^{12}+15a^{11}+13a^{10}+a^{9}-2a^{8}-18a^{7}-24a^{6}-21a^{5}-6a^{4}+9a^{3}+17a^{2}+35a+17$, $5a^{30}-6a^{29}-2a^{28}+5a^{26}-5a^{24}-3a^{23}+4a^{22}+9a^{21}-a^{20}-7a^{19}-11a^{18}+5a^{17}+9a^{16}+9a^{15}-8a^{14}-9a^{13}-a^{12}+10a^{11}+4a^{10}-10a^{9}-10a^{8}+5a^{7}+23a^{6}+6a^{5}-17a^{4}-32a^{3}+4a^{2}+27a+1$, $28a^{30}+20a^{29}+8a^{28}-18a^{27}-21a^{26}-20a^{25}-9a^{24}+15a^{23}+38a^{22}+26a^{21}-11a^{20}-38a^{19}-41a^{18}-32a^{17}-4a^{16}+40a^{15}+59a^{14}+19a^{13}-41a^{12}-68a^{11}-60a^{10}-33a^{9}+29a^{8}+97a^{7}+91a^{6}-a^{5}-82a^{4}-98a^{3}-74a^{2}-9a-28$, $9a^{30}-5a^{29}-10a^{28}-14a^{27}-3a^{26}+15a^{25}+6a^{24}+13a^{23}-2a^{21}-23a^{20}-24a^{19}+11a^{18}+20a^{17}+26a^{16}+2a^{14}-19a^{13}-36a^{12}-18a^{11}+11a^{10}+53a^{9}+24a^{8}+13a^{7}-26a^{6}-50a^{5}-37a^{4}-12a^{3}+53a^{2}+49a+10$, $270a^{30}-163a^{29}+122a^{28}-30a^{27}+52a^{26}-20a^{25}+12a^{24}-5a^{23}+24a^{22}+4a^{21}-17a^{20}-47a^{19}-58a^{18}-26a^{17}+4a^{16}+25a^{15}+18a^{14}-8a^{13}+5a^{12}+38a^{11}+82a^{10}+91a^{9}+29a^{8}-36a^{7}-87a^{6}-72a^{5}-19a^{4}-19a^{3}-49a^{2}-115a-1462$, $59a^{30}+54a^{29}-98a^{28}-3a^{27}+109a^{26}-61a^{25}-81a^{24}+118a^{23}+15a^{22}-145a^{21}+74a^{20}+122a^{19}-156a^{18}-44a^{17}+197a^{16}-71a^{15}-171a^{14}+187a^{13}+68a^{12}-256a^{11}+86a^{10}+241a^{9}-240a^{8}-128a^{7}+334a^{6}-62a^{5}-318a^{4}+271a^{3}+179a^{2}-422a-238$, $4a^{30}+9a^{29}-6a^{28}-8a^{27}+16a^{26}-8a^{25}-9a^{24}+16a^{23}-3a^{22}-13a^{21}+7a^{20}+9a^{19}-6a^{18}-10a^{17}+13a^{16}+4a^{15}-23a^{14}+17a^{13}+19a^{12}-38a^{11}+3a^{10}+36a^{9}-25a^{8}-13a^{7}+28a^{6}-6a^{5}-19a^{4}+13a^{3}+22a^{2}-29a-47$, $2a^{30}+2a^{29}+12a^{28}-a^{26}-12a^{25}+a^{24}+2a^{23}+14a^{22}-3a^{20}-19a^{19}-5a^{18}+18a^{16}-9a^{14}-30a^{13}-10a^{12}+24a^{10}+3a^{9}-9a^{8}-37a^{7}-13a^{6}+6a^{5}+42a^{4}+19a^{3}-3a^{2}-43a-26$, $24a^{30}+39a^{29}+38a^{28}+24a^{27}+40a^{26}+16a^{25}+17a^{24}+10a^{23}-14a^{22}-8a^{21}-43a^{20}-40a^{19}-58a^{18}-87a^{17}-65a^{16}-107a^{15}-98a^{14}-85a^{13}-114a^{12}-69a^{11}-77a^{10}-49a^{9}-14a^{8}-5a^{7}+66a^{6}+67a^{5}+128a^{4}+179a^{3}+170a^{2}+265a+122$, $7a^{30}+10a^{29}+20a^{28}+19a^{27}-11a^{26}-10a^{25}-40a^{24}-20a^{23}-3a^{22}+12a^{21}+34a^{20}+13a^{19}-5a^{18}-14a^{17}-36a^{16}+14a^{15}+21a^{14}+59a^{13}+49a^{12}-7a^{11}-39a^{10}-88a^{9}-76a^{8}-10a^{7}+15a^{6}+93a^{5}+44a^{4}+4a^{3}-44a^{2}-83a-32$, $9a^{30}+15a^{29}+12a^{28}+10a^{27}+15a^{26}+12a^{25}+16a^{24}+18a^{23}+19a^{22}+22a^{21}+18a^{20}+21a^{19}+26a^{18}+18a^{17}+28a^{16}+37a^{15}+29a^{14}+33a^{13}+39a^{12}+32a^{11}+38a^{10}+44a^{9}+48a^{8}+55a^{7}+56a^{6}+61a^{5}+60a^{4}+50a^{3}+76a^{2}+82a+26$, $4978a^{30}-2992a^{29}+1791a^{28}-1078a^{27}+645a^{26}-386a^{25}+236a^{24}-133a^{23}+91a^{22}-43a^{21}+35a^{20}-16a^{19}+10a^{18}-12a^{17}-5a^{16}-14a^{15}-11a^{14}-12a^{13}-9a^{12}-6a^{11}-a^{10}+6a^{9}+15a^{8}+19a^{7}+20a^{6}+17a^{5}+10a^{4}+3a^{3}-7a^{2}-16a-24913$
| 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: | \( 1258080429033531000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 1258080429033531000000 \cdot 1}{2\cdot\sqrt{958751781772834738039164297352952377530037676210703017156038840281}}\cr\approx \mathstrut & 0.768127763566694 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 5*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^31 - 5*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^31 - 5*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^31 - 5*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 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 | $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/17.11.0.1}{11} }$ | $16{,}\,15$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $21{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.3.0.1}{3} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | $21{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 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 | $[3/2, 3/2]_{2}^{4}$ | |
| 3.12.12.10 | $x^{12} - 24 x^{11} + 306 x^{10} - 2004 x^{9} + 7236 x^{8} - 4374 x^{7} - 1458 x^{6} + 5832 x^{5} - 1836 x^{3} + 324 x^{2} + 486 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.11.0.1 | $x^{11} + 3 x + 11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
| 13.13.0.1 | $x^{13} + 12 x + 11$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(16690100963\)
| $\Q_{16690100963}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{16690100963}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
|
\(214\!\cdots\!951\)
| $\Q_{21\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |