Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 2)
gp: K = bnfinit(y^31 - 5*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 5*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 5*x - 2)
\( x^{31} - 5x - 2 \)
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: |
\(958755296146092547177160435809572144648164966369654411826462457856\)
\(\medspace = 2^{31}\cdot 15619263033943\cdot 55942004669294094613\cdot 510951109592180719533133\)
| 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: |
\(2\), \(15619263033943\), \(55942004669294094613\), \(510951109592180719533133\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{89291\!\cdots\!43694}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$4a^{30}-3a^{29}+4a^{27}-6a^{26}+4a^{25}-a^{24}-a^{23}+6a^{22}-10a^{21}+7a^{20}-3a^{19}+4a^{17}-8a^{16}+7a^{15}-6a^{13}+9a^{12}-11a^{11}+12a^{10}-6a^{9}-7a^{8}+10a^{7}-9a^{6}+12a^{5}-5a^{4}-9a^{3}+17a^{2}-15a-9$, $186a^{30}-82a^{29}+18a^{28}-22a^{27}+2a^{26}+6a^{25}+14a^{24}+12a^{23}+4a^{22}-6a^{21}-9a^{20}-4a^{19}+9a^{18}+21a^{17}+23a^{16}+15a^{15}-2a^{14}-14a^{13}-14a^{12}-2a^{11}+17a^{10}+24a^{9}+15a^{8}-11a^{7}-40a^{6}-51a^{5}-39a^{4}-6a^{3}+26a^{2}+32a-921$, $7a^{30}-6a^{28}+4a^{27}+a^{26}-4a^{25}+8a^{24}-3a^{23}-10a^{22}+9a^{21}+a^{20}-6a^{19}+4a^{18}-3a^{17}-9a^{16}+14a^{15}+3a^{14}-14a^{13}+6a^{12}+4a^{11}-9a^{10}+16a^{9}-3a^{8}-24a^{7}+16a^{6}+7a^{5}-13a^{4}+7a^{3}-4a^{2}-21a-7$, $12a^{30}-4a^{29}+14a^{28}+8a^{27}+4a^{26}-18a^{25}+16a^{24}-2a^{23}+34a^{22}-21a^{21}+10a^{20}-21a^{19}+44a^{18}-a^{17}+17a^{16}-26a^{15}+7a^{14}+26a^{13}+25a^{12}+12a^{11}-39a^{10}+28a^{9}+3a^{8}+75a^{7}-32a^{6}+12a^{5}-42a^{4}+89a^{3}+18a^{2}+36a-111$, $50a^{30}-12a^{29}-52a^{28}+45a^{27}+24a^{26}-60a^{25}+28a^{24}+46a^{23}-72a^{22}-7a^{21}+77a^{20}-41a^{19}-44a^{18}+76a^{17}+a^{16}-80a^{15}+28a^{14}+57a^{13}-45a^{12}-27a^{11}+61a^{10}-a^{9}-76a^{8}+45a^{7}+73a^{6}-114a^{5}-30a^{4}+171a^{3}-70a^{2}-153a-47$, $8a^{30}+15a^{29}+12a^{28}-41a^{26}+10a^{25}+21a^{24}+15a^{23}+2a^{22}-58a^{21}+15a^{20}+25a^{19}+27a^{18}-2a^{17}-74a^{16}+12a^{15}+36a^{14}+41a^{13}-3a^{12}-94a^{11}+2a^{10}+53a^{9}+54a^{8}+5a^{7}-124a^{6}-7a^{5}+67a^{4}+73a^{3}+12a^{2}-156a-61$, $11a^{30}-8a^{29}-15a^{28}-5a^{27}+18a^{26}+13a^{25}-7a^{24}-21a^{23}-6a^{22}+17a^{21}+21a^{20}-5a^{19}-30a^{18}-10a^{17}+26a^{16}+27a^{15}-16a^{14}-31a^{13}-8a^{12}+29a^{11}+29a^{10}-9a^{9}-46a^{8}-21a^{7}+46a^{6}+50a^{5}-17a^{4}-73a^{3}-21a^{2}+54a+21$, $23a^{30}-40a^{29}+7a^{27}+26a^{26}+a^{25}-36a^{24}-a^{23}-5a^{22}+63a^{21}-29a^{20}-2a^{19}-63a^{18}+67a^{17}+6a^{16}+31a^{15}-72a^{14}-14a^{13}+9a^{12}+60a^{11}+24a^{10}-87a^{9}-26a^{8}+a^{7}+140a^{6}-50a^{5}-14a^{4}-129a^{3}+107a^{2}+43a-29$, $4a^{30}+5a^{28}-a^{27}+2a^{26}-5a^{25}-a^{24}-7a^{23}+a^{22}-2a^{21}+7a^{20}+2a^{19}+8a^{18}-2a^{17}+3a^{16}-7a^{15}-2a^{14}-11a^{13}-5a^{11}+10a^{10}+6a^{9}+16a^{8}+3a^{6}-15a^{5}-7a^{4}-18a^{3}-4a-3$, $59a^{30}+89a^{29}+124a^{28}+69a^{27}+113a^{26}+133a^{25}+112a^{24}+118a^{23}+188a^{22}+128a^{21}+146a^{20}+229a^{19}+130a^{18}+194a^{17}+237a^{16}+191a^{15}+217a^{14}+308a^{13}+253a^{12}+251a^{11}+407a^{10}+262a^{9}+331a^{8}+437a^{7}+331a^{6}+395a^{5}+515a^{4}+466a^{3}+446a^{2}+695a+227$, $94a^{30}-60a^{29}-9a^{28}-39a^{27}-10a^{26}-32a^{25}-40a^{24}-31a^{23}-31a^{22}-36a^{21}-56a^{20}-41a^{19}-29a^{18}-65a^{17}-55a^{16}-48a^{15}-45a^{14}-66a^{13}-80a^{12}-35a^{11}-64a^{10}-90a^{9}-69a^{8}-62a^{7}-71a^{6}-117a^{5}-91a^{4}-58a^{3}-135a^{2}-133a-585$, $36a^{30}+18a^{29}-12a^{28}-30a^{27}+11a^{26}+50a^{25}-58a^{24}-12a^{23}+61a^{22}-7a^{21}-36a^{20}-38a^{19}+82a^{18}+14a^{17}-98a^{16}+48a^{15}+44a^{14}-32a^{13}-61a^{12}+28a^{11}+132a^{10}-118a^{9}-76a^{8}+114a^{7}+23a^{6}-58a^{5}-74a^{4}+150a^{3}+35a^{2}-246a-93$, $152a^{30}+338a^{29}+189a^{28}-177a^{27}-404a^{26}-230a^{25}+210a^{24}+488a^{23}+283a^{22}-249a^{21}-593a^{20}-355a^{19}+289a^{18}+721a^{17}+453a^{16}-325a^{15}-873a^{14}-581a^{13}+354a^{12}+1047a^{11}+737a^{10}-376a^{9}-1240a^{8}-917a^{7}+394a^{6}+1453a^{5}+1119a^{4}-415a^{3}-1697a^{2}-1349a-311$, $180a^{30}+87a^{29}+122a^{28}+269a^{27}+147a^{26}+157a^{25}+304a^{24}+137a^{23}+111a^{22}+253a^{21}+28a^{20}-41a^{19}+103a^{18}-170a^{17}-272a^{16}-106a^{15}-409a^{14}-519a^{13}-284a^{12}-583a^{11}-669a^{10}-326a^{9}-583a^{8}-610a^{7}-122a^{6}-325a^{5}-299a^{4}+345a^{3}+185a^{2}+232a+89$, $20a^{30}-31a^{29}+12a^{28}-32a^{27}+20a^{26}-10a^{25}+28a^{24}-4a^{23}-3a^{22}-13a^{21}-25a^{20}+8a^{19}-15a^{18}+53a^{17}-12a^{16}+43a^{15}-54a^{14}+20a^{13}-58a^{12}+39a^{11}-2a^{10}+66a^{9}+11a^{8}-a^{7}-26a^{6}-58a^{5}+9a^{4}-28a^{3}+102a^{2}-11a-23$, $8a^{30}-13a^{29}+16a^{28}-18a^{27}+6a^{26}-11a^{25}+6a^{24}-10a^{23}+30a^{22}-7a^{21}+20a^{20}-3a^{19}+3a^{18}-29a^{17}+19a^{16}-29a^{15}+12a^{14}-6a^{13}+6a^{12}-39a^{11}+24a^{10}-34a^{9}+22a^{8}+17a^{7}+48a^{6}-10a^{5}+51a^{4}-44a^{3}-10a^{2}-31a-31$
| 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: | \( 4907974822306732000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 4907974822306732000000 \cdot 1}{2\cdot\sqrt{958755296146092547177160435809572144648164966369654411826462457856}}\cr\approx \mathstrut & 2.99658489823367 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 2)
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 - 2, 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 - 2);
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 - 2);
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 | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $16{,}\,15$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $30{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
| 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
|
\(15619263033943\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ | ||
|
\(55942004669294094613\)
| $\Q_{55942004669294094613}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(510\!\cdots\!133\)
| $\Q_{51\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{51\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |