Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 3*x - 1)
gp: K = bnfinit(y^31 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 3*x - 1)
\( x^{31} - 3x - 1 \)
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: |
\(127173474825631541368752576367041389356970940685243955265569\)
\(\medspace = 7\cdot 17\cdot 23\cdot 15726392777460683\cdot 29\!\cdots\!39\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(80.65\) | 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: | $7^{1/2}17^{1/2}23^{1/2}15726392777460683^{1/2}2954558696113433838143509264321373433139^{1/2}\approx 3.5661390161578326e+29$ | ||
| Ramified primes: |
\(7\), \(17\), \(23\), \(15726392777460683\), \(29545\!\cdots\!33139\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{12717\!\cdots\!65569}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{30}-3$, $a+1$, $a^{16}-2a$, $a^{30}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}-1$, $a^{23}-a^{21}-a^{18}-a^{15}-a^{14}+a^{11}+a^{8}-a^{6}+a^{5}-3a^{3}-2a^{2}-1$, $a^{30}-2a^{29}+a^{28}+2a^{27}-2a^{26}+2a^{25}-3a^{24}+3a^{23}-3a^{22}+a^{21}+2a^{20}-a^{19}+2a^{18}-4a^{17}+4a^{16}-4a^{15}+3a^{11}-5a^{10}+5a^{9}-4a^{8}+a^{7}-2a^{6}+2a^{5}+6a^{4}-5a^{3}+5a^{2}-5a-1$, $2a^{30}-3a^{29}+5a^{28}-2a^{26}+7a^{25}-3a^{24}+a^{23}+6a^{22}-4a^{21}+4a^{20}+4a^{19}-4a^{18}+7a^{17}-a^{15}+8a^{14}-3a^{13}+3a^{12}+8a^{11}-6a^{10}+7a^{9}+5a^{8}-6a^{7}+11a^{6}+a^{5}-2a^{4}+13a^{3}-4a^{2}+4a+4$, $3a^{29}+2a^{28}+3a^{26}+2a^{25}+3a^{23}+3a^{22}+4a^{20}+3a^{19}+5a^{17}+3a^{16}+6a^{14}+3a^{13}+6a^{11}+4a^{10}+a^{9}+7a^{8}+5a^{7}+2a^{6}+7a^{5}+4a^{4}+2a^{3}+7a^{2}+6a+3$, $2a^{30}-3a^{29}-2a^{28}+8a^{27}-5a^{26}-4a^{25}+7a^{24}-2a^{23}-3a^{22}+2a^{21}+5a^{20}-6a^{19}-4a^{18}+11a^{17}-5a^{16}-5a^{15}+9a^{14}-2a^{13}-7a^{12}+3a^{11}+8a^{10}-6a^{9}-6a^{8}+13a^{7}-6a^{6}-10a^{5}+15a^{4}-10a^{2}+2a+3$, $2a^{30}-2a^{29}-3a^{28}+4a^{26}+3a^{25}-3a^{24}-4a^{23}-a^{22}+5a^{21}+4a^{20}-2a^{19}-5a^{18}-2a^{17}+4a^{16}+5a^{15}-2a^{14}-5a^{13}-3a^{12}+5a^{11}+5a^{10}-3a^{9}-7a^{8}-2a^{7}+6a^{6}+6a^{5}-4a^{4}-7a^{3}-3a^{2}+8a+2$, $7a^{30}+a^{29}-9a^{28}-9a^{27}+2a^{26}+8a^{25}+3a^{24}-2a^{23}+2a^{22}+3a^{21}-7a^{20}-13a^{19}-a^{18}+14a^{17}+13a^{16}-3a^{15}-9a^{14}-a^{13}-9a^{11}-5a^{10}+14a^{9}+22a^{8}+2a^{7}-21a^{6}-16a^{5}+4a^{4}+5a^{3}-3a^{2}+5a+1$, $4a^{30}+3a^{29}+a^{28}-3a^{27}-6a^{26}-a^{25}+2a^{23}+7a^{22}+3a^{21}-2a^{20}-5a^{19}-6a^{18}-4a^{17}+5a^{16}+7a^{15}+4a^{14}+5a^{13}-5a^{12}-11a^{11}-7a^{10}+7a^{8}+13a^{7}+7a^{6}-6a^{5}-6a^{4}-12a^{3}-10a^{2}+6a+2$, $a^{30}+2a^{29}+a^{28}+a^{27}-a^{26}-a^{25}-a^{24}+2a^{22}+2a^{21}+2a^{20}-2a^{18}-2a^{17}-2a^{16}+2a^{15}+3a^{14}+2a^{13}+a^{12}-3a^{11}-4a^{10}-3a^{9}-2a^{8}+a^{7}+2a^{6}+3a^{5}-a^{4}-5a^{3}-5a^{2}-4a$, $4a^{30}-a^{29}-2a^{28}+3a^{27}+4a^{26}-2a^{25}-a^{24}+4a^{23}+3a^{22}-2a^{21}+5a^{19}+a^{18}-a^{17}+2a^{16}+5a^{15}-a^{14}-a^{13}+6a^{12}+4a^{11}-3a^{10}+9a^{8}+4a^{7}-6a^{6}+2a^{5}+11a^{4}+4a^{3}-8a^{2}+5a+1$, $a^{30}+a^{28}-6a^{27}+a^{25}+5a^{24}+a^{23}-3a^{22}+a^{21}-3a^{20}-a^{19}-4a^{18}+4a^{17}+8a^{16}+a^{15}-a^{14}-9a^{13}+a^{12}-4a^{11}+a^{10}+3a^{9}+6a^{8}+9a^{7}-8a^{6}-5a^{5}-11a^{4}+6a^{3}+3a^{2}+3a+2$, $9a^{30}+14a^{29}+13a^{28}+12a^{27}+9a^{26}+12a^{25}+12a^{24}+7a^{23}+a^{21}+a^{20}-7a^{19}-13a^{18}-17a^{17}-15a^{16}-18a^{15}-25a^{14}-30a^{13}-21a^{12}-20a^{11}-23a^{10}-24a^{9}-13a^{8}-4a^{7}+8a^{4}+28a^{3}+32a^{2}+29a+8$
| 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: | \( 227786011641746100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 227786011641746100 \cdot 1}{2\cdot\sqrt{127173474825631541368752576367041389356970940685243955265569}}\cr\approx \mathstrut & 0.381862428249440 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 3*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 - 3*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 - 3*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 - 3*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 | $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $30{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $31$ | R | $19{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.13.0.1}{13} }$ | R | $22{,}\,{\href{/padicField/19.9.0.1}{9} }$ | R | $20{,}\,{\href{/padicField/29.11.0.1}{11} }$ | $30{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $17{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.9.0.1 | $x^{9} + 6 x^{4} + x^{3} + 6 x + 4$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 7.17.0.1 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.9.0.1 | $x^{9} + 7 x^{2} + 8 x + 14$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 17.14.0.1 | $x^{14} + x^{8} + 11 x^{7} + x^{6} + 8 x^{5} + 16 x^{4} + 13 x^{3} + 9 x^{2} + 3 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.18.0.1 | $x^{18} + x^{12} + 18 x^{11} + 2 x^{10} + x^{9} + 18 x^{8} + 3 x^{7} + 16 x^{6} + 21 x^{5} + 11 x^{3} + 3 x^{2} + 19 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(15726392777460683\)
| $\Q_{15726392777460683}$ | $x$ | $1$ | $1$ | $0$ | 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 $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(295\!\cdots\!139\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |