Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 3*x - 3)
gp: K = bnfinit(y^31 - 3*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 3*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 3*x - 3)
\( x^{31} - 3x - 3 \)
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: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3387218110869654681977390701144469962323789296982843961159719\)
\(\medspace = -\,3^{30}\cdot 7\cdot 2713\cdot 14387\cdot 3413878991\cdot 72413150415649\cdot 243568924508677\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(89.66\) | 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: | $3^{30/31}7^{1/2}2713^{1/2}14387^{1/2}3413878991^{1/2}72413150415649^{1/2}243568924508677^{1/2}\approx 3.7139239555445536e+23$ | ||
| Ramified primes: |
\(3\), \(7\), \(2713\), \(14387\), \(3413878991\), \(72413150415649\), \(243568924508677\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-16451\!\cdots\!34431}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: |
$a+1$, $a^{16}-2a-1$, $a^{29}+a^{28}-2a^{27}+a^{25}+a^{24}-3a^{23}+a^{22}+a^{21}+a^{20}-4a^{19}+2a^{18}+2a^{17}-a^{16}-3a^{15}+2a^{14}+2a^{13}-2a^{12}-2a^{11}+3a^{10}+a^{9}-2a^{8}-2a^{7}+5a^{6}-5a^{4}+6a^{2}-a-7$, $6a^{30}-11a^{29}+8a^{28}-a^{27}-a^{26}-a^{25}-3a^{24}+11a^{23}-12a^{22}+4a^{21}+3a^{19}-2a^{18}-9a^{17}+15a^{16}-8a^{15}-4a^{13}+7a^{12}+4a^{11}-17a^{10}+12a^{9}+2a^{7}-12a^{6}+4a^{5}+15a^{4}-17a^{3}+a^{2}+a-4$, $20a^{30}-20a^{29}+18a^{28}-17a^{27}+15a^{26}-13a^{25}+13a^{24}-12a^{23}+12a^{22}-11a^{21}+8a^{20}-8a^{19}+5a^{18}-6a^{17}+5a^{16}-4a^{15}+5a^{14}-2a^{13}+2a^{12}-a^{10}-a^{9}-a^{8}+3a^{5}-a^{4}+7a^{3}-2a^{2}+2a-62$, $a^{30}-a^{29}+a^{28}-a^{27}-a^{26}+a^{25}-a^{24}+3a^{23}-4a^{22}+3a^{21}-a^{20}+2a^{19}-2a^{17}-3a^{15}+4a^{14}-a^{12}-a^{10}+4a^{9}-2a^{8}+2a^{7}-4a^{6}-2a^{5}+3a^{4}+2a^{2}-4a+1$, $2a^{30}-5a^{29}-a^{28}+8a^{27}-5a^{26}-4a^{25}+7a^{24}-7a^{22}+3a^{21}+6a^{20}-7a^{19}-3a^{18}+8a^{17}-a^{16}-10a^{15}+8a^{14}+5a^{13}-10a^{12}-2a^{11}+14a^{10}-7a^{9}-10a^{8}+14a^{7}+4a^{6}-20a^{5}+10a^{4}+14a^{3}-17a^{2}-5a+17$, $4a^{30}+4a^{29}-9a^{28}+9a^{27}-2a^{26}-7a^{25}+13a^{24}-6a^{23}-5a^{22}+10a^{21}-9a^{20}+4a^{19}+3a^{18}-16a^{17}+14a^{16}+4a^{15}-17a^{14}+12a^{13}-a^{12}-2a^{11}+13a^{10}-19a^{9}+5a^{8}+19a^{7}-22a^{6}+6a^{5}+6a^{4}-17a^{3}+19a^{2}-9a-28$, $6a^{30}-3a^{29}+2a^{28}+3a^{27}-3a^{26}+8a^{25}-11a^{24}+11a^{23}-16a^{22}+13a^{21}-16a^{20}+15a^{19}-10a^{18}+11a^{17}-a^{16}+6a^{14}-13a^{13}+11a^{12}-19a^{11}+15a^{10}-15a^{9}+17a^{8}-5a^{7}+10a^{6}+2a^{5}-2a^{4}-12a^{2}+2a-32$, $2a^{28}-a^{26}-2a^{25}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}-2a^{18}-3a^{17}-a^{16}+a^{15}+2a^{14}+a^{13}-a^{11}-a^{10}-2a^{9}-2a^{8}+3a^{7}+5a^{6}+4a^{5}-a^{4}-4a^{3}-4a^{2}-2a-1$, $a^{30}-16a^{29}-3a^{28}+13a^{27}-2a^{26}-16a^{25}+5a^{24}+19a^{23}-5a^{22}-14a^{21}+15a^{20}+23a^{19}-9a^{18}-10a^{17}+25a^{16}+21a^{15}-18a^{14}-7a^{13}+32a^{12}+12a^{11}-31a^{10}-7a^{9}+32a^{8}-6a^{7}-46a^{6}-4a^{5}+29a^{4}-29a^{3}-59a^{2}+6a+26$, $16a^{30}-17a^{29}+2a^{28}+19a^{27}-20a^{26}+11a^{25}+3a^{24}-24a^{23}+23a^{22}-a^{21}-13a^{20}+25a^{19}-23a^{18}-8a^{17}+27a^{16}-23a^{15}+16a^{14}+14a^{13}-39a^{12}+21a^{11}+a^{10}-20a^{9}+44a^{8}-21a^{7}-22a^{6}+30a^{5}-36a^{4}+18a^{3}+39a^{2}-46a-29$, $5a^{30}-4a^{29}-3a^{28}+3a^{27}+3a^{26}-2a^{25}-5a^{24}+4a^{23}+5a^{22}-7a^{21}+4a^{19}+2a^{18}-7a^{17}+7a^{15}+a^{14}-13a^{13}+6a^{12}+8a^{11}-4a^{10}-6a^{9}-2a^{8}+18a^{7}-14a^{6}-4a^{5}+9a^{4}+3a^{3}-6a^{2}-12a+5$, $3a^{30}-4a^{29}+5a^{28}-5a^{27}+4a^{26}-a^{25}-3a^{24}+5a^{23}-5a^{22}+3a^{21}-a^{20}+a^{19}-2a^{18}+a^{17}+a^{16}-4a^{15}+6a^{14}-5a^{13}+2a^{12}-a^{11}+a^{10}+a^{9}-3a^{8}+7a^{7}-11a^{6}+8a^{5}-5a^{3}+10a^{2}-9a-5$, $a^{30}-a^{28}+a^{27}-a^{25}+a^{23}-a^{19}+2a^{17}-a^{16}-3a^{15}+2a^{14}+3a^{13}-2a^{12}-3a^{11}+3a^{10}-3a^{8}+a^{7}+3a^{6}-a^{5}-a^{4}+2a^{3}-3a^{2}-2a+1$
| 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: | \( 1235026277671475000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 1235026277671475000 \cdot 1}{2\cdot\sqrt{3387218110869654681977390701144469962323789296982843961159719}}\cr\approx \mathstrut & 0.630162625881638 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - 3*x - 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 - 3*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 - 3*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 - 3*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 | $20{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | $19{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.13.0.1}{13} }$ | $20{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $31$ | $19{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $26{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $16{,}\,15$ | $27{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $31$ | $31$ | $1$ | $30$ | |||
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 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 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(2713\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(14387\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(3413878991\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(72413150415649\)
| $\Q_{72413150415649}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(243568924508677\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |