Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 4*x - 1)
gp: K = bnfinit(y^31 + 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 4*x - 1)
\( x^{31} + 4x - 1 \)
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: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-949505255199079242384975296448931958610643029059314756044734431\)
\(\medspace = -\,7\cdot 13\!\cdots\!33\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(107.53\) | 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: | $7^{1/2}135643607885582748912139328064133136944377575579902108006390633^{1/2}\approx 3.0814043149172737e+31$ | ||
| Ramified primes: |
\(7\), \(13564\!\cdots\!90633\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-94950\!\cdots\!34431}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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
$C_{2}$, which has order $2$ (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: |
$a$, $9a^{29}-11a^{28}+5a^{27}+3a^{26}-6a^{25}+a^{24}+a^{23}+3a^{22}-11a^{21}+13a^{20}-2a^{19}-12a^{18}+19a^{17}-10a^{16}-2a^{15}+3a^{14}+3a^{13}-7a^{12}-5a^{11}+22a^{10}-28a^{9}+13a^{8}+14a^{7}-24a^{6}+15a^{5}+3a^{3}-20a^{2}+23a-3$, $5a^{30}+5a^{29}+3a^{28}+4a^{27}-3a^{24}-4a^{23}-8a^{22}-9a^{21}-5a^{20}-5a^{19}-4a^{18}-7a^{17}+a^{15}+6a^{14}+5a^{13}+9a^{12}+11a^{11}+9a^{10}+12a^{9}+8a^{8}+9a^{7}-6a^{6}-5a^{5}-7a^{4}-4a^{3}-13a^{2}-18a+6$, $6a^{29}+8a^{28}+12a^{27}+14a^{26}+12a^{25}+13a^{24}+12a^{23}+7a^{22}+6a^{21}-a^{20}-9a^{19}-12a^{18}-18a^{17}-21a^{16}-17a^{15}-19a^{14}-17a^{13}-10a^{12}-9a^{11}+2a^{10}+14a^{9}+18a^{8}+27a^{7}+30a^{6}+23a^{5}+28a^{4}+24a^{3}+13a^{2}+12a-5$, $9a^{30}-2a^{29}-14a^{28}-15a^{27}-a^{26}+18a^{25}+20a^{24}-18a^{22}-17a^{21}-4a^{20}+10a^{19}+17a^{18}+12a^{17}-14a^{15}-25a^{14}-14a^{13}+20a^{12}+38a^{11}+16a^{10}-22a^{9}-40a^{8}-21a^{7}+19a^{6}+37a^{5}+20a^{4}-2a^{3}-20a^{2}-37a+9$, $a^{30}-7a^{29}+7a^{28}-a^{27}+12a^{26}-16a^{25}+10a^{24}-12a^{23}+25a^{22}-21a^{21}+16a^{20}-26a^{19}+33a^{18}-25a^{17}+24a^{16}-39a^{15}+36a^{14}-25a^{13}+32a^{12}-46a^{11}+26a^{10}-24a^{9}+38a^{8}-36a^{7}+10a^{6}-23a^{5}+27a^{4}-10a^{3}-11a^{2}-10a+4$, $9a^{30}-14a^{29}-6a^{28}+4a^{27}+11a^{26}+4a^{25}-22a^{24}-a^{23}+23a^{22}-6a^{21}-11a^{20}+3a^{19}-3a^{18}+11a^{17}+2a^{16}-17a^{15}+9a^{14}-8a^{12}+22a^{11}-11a^{10}-29a^{9}+26a^{8}+16a^{7}-11a^{6}-13a^{5}-24a^{4}+42a^{3}+37a^{2}-72a+14$, $13a^{30}+13a^{29}+a^{28}-3a^{27}-14a^{26}-19a^{25}-15a^{24}-18a^{23}+2a^{22}+a^{21}+23a^{20}+18a^{19}+26a^{18}+15a^{17}+2a^{16}-5a^{15}-29a^{14}-22a^{13}-34a^{12}-16a^{11}-5a^{10}+13a^{9}+30a^{8}+35a^{7}+38a^{6}+19a^{5}+9a^{4}-24a^{3}-31a^{2}-53a+17$, $10a^{30}-8a^{29}+5a^{28}+2a^{27}-10a^{26}+10a^{25}-7a^{24}+6a^{23}-a^{22}-3a^{21}+2a^{20}-5a^{19}+11a^{18}-10a^{17}+7a^{16}-4a^{15}-8a^{14}+17a^{13}-14a^{12}+11a^{11}-7a^{10}-5a^{9}+9a^{8}-7a^{7}+16a^{6}-23a^{5}+18a^{4}-13a^{3}-a^{2}+33a-10$, $16a^{30}+13a^{29}+10a^{28}+13a^{27}+15a^{26}+7a^{25}+12a^{24}+13a^{23}+4a^{22}+5a^{21}+8a^{20}-4a^{18}+6a^{17}-7a^{16}-9a^{15}-3a^{14}-13a^{13}-23a^{12}-12a^{11}-19a^{10}-35a^{9}-16a^{8}-25a^{7}-38a^{6}-30a^{5}-26a^{4}-50a^{3}-37a^{2}-28a+12$, $3a^{30}-3a^{29}-6a^{28}-15a^{27}-14a^{26}-7a^{25}-3a^{24}+5a^{23}+9a^{22}+20a^{21}+17a^{20}+8a^{19}+4a^{18}-7a^{17}-12a^{16}-27a^{15}-22a^{14}-10a^{13}-5a^{12}+8a^{11}+14a^{10}+35a^{9}+27a^{8}+11a^{7}+5a^{6}-10a^{5}-17a^{4}-44a^{3}-34a^{2}-13a+8$, $3a^{30}+13a^{29}+9a^{28}-31a^{27}+a^{26}+21a^{25}+12a^{24}-19a^{23}-26a^{22}+32a^{21}+6a^{20}-4a^{19}-14a^{18}-12a^{17}+22a^{16}+4a^{15}+15a^{14}-47a^{13}-11a^{12}+64a^{11}-40a^{9}-41a^{8}+71a^{7}+19a^{6}-50a^{5}+8a^{4}-28a^{3}+47a^{2}+17a-6$, $9a^{30}+11a^{29}+5a^{28}-4a^{27}-9a^{26}-6a^{25}+3a^{24}+12a^{23}+15a^{22}+10a^{21}-10a^{19}-16a^{18}-17a^{17}-14a^{16}-7a^{15}+4a^{14}+17a^{13}+27a^{12}+26a^{11}+10a^{10}-16a^{9}-39a^{8}-43a^{7}-22a^{6}+14a^{5}+45a^{4}+51a^{3}+28a^{2}-9a$, $20a^{30}-25a^{29}-2a^{28}+25a^{27}-10a^{26}-14a^{25}+13a^{24}+4a^{23}-19a^{22}+15a^{21}+14a^{20}-33a^{19}+14a^{18}+14a^{17}-28a^{16}+30a^{15}-2a^{14}-39a^{13}+36a^{12}+6a^{11}-33a^{10}+23a^{9}+18a^{8}-46a^{7}-4a^{6}+77a^{5}-46a^{4}-60a^{3}+104a^{2}-38a+3$, $5a^{30}-24a^{29}+32a^{28}-17a^{27}-7a^{26}+37a^{25}-33a^{24}+19a^{23}+11a^{22}-25a^{21}+33a^{20}-26a^{19}+10a^{18}+10a^{17}-40a^{16}+51a^{15}-42a^{14}-15a^{13}+59a^{12}-81a^{11}+41a^{10}+16a^{9}-68a^{8}+79a^{7}-40a^{6}+7a^{5}+44a^{4}-61a^{3}+92a^{2}-56a+10$
| 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: | \( 20750959184225810000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 20750959184225810000 \cdot 2}{2\cdot\sqrt{949505255199079242384975296448931958610643029059314756044734431}}\cr\approx \mathstrut & 1.26478734288511 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 + 4*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 + 4*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 + 4*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 + 4*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 $S_{31}$ 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 | ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 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 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(135\!\cdots\!633\)
| $\Q_{13\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{13\!\cdots\!33}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |