Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 1)
gp: K = bnfinit(y^31 - 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 5*x - 1)
\( x^{31} - 5x - 1 \)
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: |
\(958755296164420433325387643496229889041389356970940685243955265569\)
\(\medspace = 13\cdot 1877\cdot 25841\cdot 202437701896105219\cdot 75\!\cdots\!11\)
| 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: | $13^{1/2}1877^{1/2}25841^{1/2}202437701896105219^{1/2}7511028960944005414303516302480340104011^{1/2}\approx 9.791605058234429e+32$ | ||
| Ramified primes: |
\(13\), \(1877\), \(25841\), \(202437701896105219\), \(75110\!\cdots\!04011\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{95875\!\cdots\!65569}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $a^{15}+2$, $2a^{29}-5a^{28}+5a^{27}-a^{26}-2a^{25}+4a^{24}-7a^{23}+7a^{22}-2a^{21}-4a^{20}+7a^{19}-8a^{18}+9a^{17}-6a^{16}-3a^{15}+11a^{14}-12a^{13}+11a^{12}-9a^{11}-a^{10}+12a^{9}-14a^{8}+12a^{7}-9a^{6}+a^{5}+12a^{4}-19a^{3}+13a^{2}-4a-1$, $9a^{30}-16a^{29}+13a^{28}-9a^{27}+11a^{26}-5a^{25}-5a^{24}+2a^{23}+2a^{22}+2a^{21}+5a^{20}-17a^{19}+12a^{18}-7a^{17}+13a^{16}-5a^{15}-8a^{14}-2a^{13}+12a^{12}-8a^{11}+23a^{10}-45a^{9}+37a^{8}-30a^{7}+42a^{6}-28a^{5}+5a^{4}-21a^{3}+33a^{2}-20a-4$, $3a^{30}+6a^{29}+5a^{28}+2a^{27}+8a^{26}+2a^{25}+5a^{24}+4a^{23}+4a^{22}+4a^{21}+2a^{20}+7a^{19}+7a^{17}+3a^{16}+5a^{15}+8a^{14}+3a^{13}+13a^{12}+5a^{11}+15a^{10}+9a^{9}+19a^{8}+14a^{7}+16a^{6}+29a^{5}+12a^{4}+31a^{3}+22a^{2}+30a+6$, $9a^{30}+3a^{29}-9a^{28}+a^{27}+a^{26}-a^{25}+5a^{24}-7a^{23}-7a^{22}+12a^{21}-10a^{19}+4a^{18}-a^{17}+a^{16}+11a^{15}-14a^{14}-6a^{13}+27a^{12}-7a^{11}-20a^{10}+24a^{9}+3a^{8}-16a^{7}+18a^{6}-7a^{5}-14a^{4}+37a^{3}-11a^{2}-45a-9$, $13a^{30}-2a^{29}-18a^{28}+20a^{27}-9a^{26}+a^{25}+7a^{24}-7a^{23}-8a^{22}+11a^{21}+4a^{20}-5a^{19}-3a^{18}-8a^{17}+13a^{16}+8a^{15}-23a^{14}+20a^{13}-7a^{12}-36a^{11}+79a^{10}-48a^{9}-24a^{8}+47a^{7}-33a^{6}+21a^{5}+8a^{4}-28a^{3}+15a^{2}-29a-8$, $20a^{30}-17a^{29}-5a^{28}+29a^{27}-25a^{26}-2a^{25}+27a^{24}-23a^{23}+3a^{22}+12a^{21}-10a^{20}+11a^{19}-16a^{18}+15a^{17}+14a^{16}-42a^{15}+38a^{14}+15a^{13}-59a^{12}+51a^{11}+12a^{10}-56a^{9}+46a^{8}+2a^{7}-23a^{6}+18a^{5}-11a^{4}+30a^{3}-19a^{2}-32a-4$, $19a^{30}-9a^{29}-9a^{27}+27a^{26}-17a^{25}-29a^{24}+77a^{23}-63a^{22}-23a^{21}+122a^{20}-130a^{19}+17a^{18}+143a^{17}-201a^{16}+88a^{15}+125a^{14}-251a^{13}+173a^{12}+63a^{11}-252a^{10}+235a^{9}-18a^{8}-200a^{7}+242a^{6}-76a^{5}-123a^{4}+191a^{3}-85a^{2}-49a-4$, $113a^{30}-100a^{29}-104a^{28}+135a^{27}+88a^{26}-169a^{25}-62a^{24}+207a^{23}+38a^{22}-232a^{21}+2a^{20}+250a^{19}-57a^{18}-260a^{17}+127a^{16}+265a^{15}-201a^{14}-259a^{13}+274a^{12}+224a^{11}-366a^{10}-185a^{9}+447a^{8}+124a^{7}-518a^{6}-51a^{5}+556a^{4}-73a^{3}-596a^{2}+226a+66$, $21a^{30}+6a^{29}-25a^{28}+51a^{27}+18a^{26}-73a^{25}+29a^{24}+47a^{23}-81a^{22}-39a^{21}+77a^{20}-58a^{19}-18a^{18}+25a^{17}+6a^{16}+40a^{15}+15a^{14}-46a^{13}+97a^{12}+60a^{11}-161a^{10}+38a^{9}+137a^{8}-209a^{7}-55a^{6}+107a^{5}-53a^{4}-108a^{3}+94a^{2}-7a-7$, $67a^{30}-21a^{29}-16a^{28}+63a^{27}-73a^{26}+61a^{25}-24a^{24}-33a^{23}+60a^{22}-85a^{21}+45a^{20}-4a^{19}-60a^{18}+97a^{17}-87a^{16}+54a^{15}+48a^{14}-109a^{13}+182a^{12}-159a^{11}+89a^{10}+28a^{9}-171a^{8}+225a^{7}-264a^{6}+142a^{5}-16a^{4}-173a^{3}+298a^{2}-311a-74$, $136a^{30}+122a^{29}+100a^{28}+70a^{27}+34a^{26}-10a^{25}-58a^{24}-109a^{23}-163a^{22}-213a^{21}-260a^{20}-301a^{19}-331a^{18}-349a^{17}-353a^{16}-337a^{15}-305a^{14}-251a^{13}-179a^{12}-92a^{11}+15a^{10}+135a^{9}+259a^{8}+390a^{7}+515a^{6}+630a^{5}+731a^{4}+802a^{3}+850a^{2}+861a+143$, $18a^{30}+10a^{29}+14a^{28}-6a^{27}+28a^{26}+23a^{25}+23a^{24}-5a^{23}+19a^{22}+27a^{21}+21a^{20}+19a^{19}+34a^{18}+42a^{17}-10a^{16}+23a^{15}+50a^{14}+72a^{13}-7a^{12}+42a^{11}+45a^{10}+42a^{9}+5a^{8}+92a^{7}+87a^{6}+9a^{5}+7a^{4}+74a^{3}+110a^{2}+25a-1$, $526a^{30}+307a^{29}-472a^{28}-674a^{27}-779a^{26}+24a^{25}+523a^{24}+1127a^{23}+592a^{22}+20a^{21}-1113a^{20}-1123a^{19}-873a^{18}+546a^{17}+1190a^{16}+1674a^{15}+431a^{14}-608a^{13}-2018a^{12}-1466a^{11}-555a^{10}+1646a^{9}+2153a^{8}+2080a^{7}-364a^{6}-1930a^{5}-3342a^{4}-1579a^{3}+395a^{2}+3461a+667$, $5623a^{30}+8092a^{29}-4907a^{28}-9339a^{27}+2150a^{26}+12349a^{25}-1175a^{24}-12217a^{23}-3490a^{22}+14791a^{21}+5449a^{20}-13504a^{19}-10693a^{18}+13659a^{17}+14498a^{16}-11763a^{15}-18432a^{14}+7700a^{13}+24127a^{12}-5324a^{11}-25251a^{10}-3338a^{9}+31126a^{8}+7074a^{7}-29134a^{6}-18490a^{5}+31616a^{4}+25172a^{3}-27322a^{2}-35621a-5885$
| 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: | \( 1341618162426672300000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 1341618162426672300000 \cdot 1}{2\cdot\sqrt{958755296164420433325387643496229889041389356970940685243955265569}}\cr\approx \mathstrut & 0.819130674104767 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 1)
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 - 1, 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 - 1);
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 - 1);
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 | $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $31$ | $31$ | ${\href{/padicField/31.3.0.1}{3} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | 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 | $x^{13} + 12 x + 11$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(1877\)
| $\Q_{1877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
|
\(25841\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(202437701896105219\)
| $\Q_{202437701896105219}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(751\!\cdots\!011\)
| $\Q_{75\!\cdots\!11}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |