Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 4*x - 2)
gp: K = bnfinit(y^31 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 4*x - 2)
\( x^{31} + 4x - 2 \)
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: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-949505273526965390612182983106676351835033630345588173537542144\)
\(\medspace = -\,2^{30}\cdot 7\cdot 195469\cdot 212031575390627\cdot 30\!\cdots\!91\)
| 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: | $2^{30/31}7^{1/2}195469^{1/2}212031575390627^{1/2}3048042651910299213783523530349391^{1/2}\approx 1.839154043245885e+27$ | ||
| Ramified primes: |
\(2\), \(7\), \(195469\), \(212031575390627\), \(30480\!\cdots\!49391\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-88429\!\cdots\!34431}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$2a^{29}-a^{28}+3a^{27}+a^{26}+3a^{24}-a^{23}+a^{22}+a^{21}-2a^{20}+3a^{19}-a^{18}-a^{17}+4a^{16}-3a^{15}+4a^{14}+a^{13}+8a^{11}-2a^{10}+4a^{9}+4a^{8}-2a^{7}+7a^{6}-a^{5}-a^{4}+6a^{3}-10a^{2}+3a-1$, $3a^{30}+12a^{29}-10a^{28}-a^{27}+16a^{26}-8a^{25}-5a^{24}+20a^{23}-4a^{22}-10a^{21}+22a^{20}-17a^{18}+21a^{17}+5a^{16}-24a^{15}+19a^{14}+13a^{13}-29a^{12}+16a^{11}+23a^{10}-34a^{9}+8a^{8}+33a^{7}-39a^{6}-6a^{5}+43a^{4}-39a^{3}-23a^{2}+55a-19$, $2a^{30}-3a^{29}+3a^{28}-3a^{27}+5a^{26}-7a^{25}+4a^{24}-2a^{22}+a^{20}+2a^{19}-2a^{18}-2a^{17}+9a^{16}-10a^{15}+5a^{14}-4a^{13}+5a^{12}-4a^{11}-4a^{10}+12a^{9}-12a^{8}+7a^{7}-2a^{6}+4a^{5}-3a^{4}-5a^{3}+11a^{2}-9a+3$, $9a^{30}-11a^{28}+8a^{27}+3a^{26}-14a^{25}+7a^{24}+8a^{23}-15a^{22}+5a^{21}+14a^{20}-16a^{19}+a^{18}+19a^{17}-17a^{16}-6a^{15}+24a^{14}-15a^{13}-15a^{12}+29a^{11}-9a^{10}-24a^{9}+32a^{8}-35a^{6}+31a^{5}+11a^{4}-44a^{3}+25a^{2}+27a-15$, $33a^{30}+22a^{29}+13a^{28}+3a^{26}+5a^{25}-4a^{24}+a^{23}+7a^{22}-5a^{21}-4a^{20}+6a^{19}-4a^{18}-3a^{17}+12a^{16}-7a^{14}+9a^{13}-a^{12}-10a^{11}+11a^{10}+5a^{9}-13a^{8}+6a^{7}+7a^{6}-16a^{5}+7a^{4}+19a^{3}-15a^{2}+149$, $40a^{30}+46a^{29}+39a^{28}+22a^{27}-3a^{26}-29a^{25}-47a^{24}-53a^{23}-44a^{22}-21a^{21}+10a^{20}+39a^{19}+59a^{18}+65a^{17}+50a^{16}+20a^{15}-15a^{14}-51a^{13}-74a^{12}-74a^{11}-55a^{10}-19a^{9}+27a^{8}+67a^{7}+89a^{6}+88a^{5}+62a^{4}+13a^{3}-41a^{2}-84a+49$, $302a^{30}+150a^{29}+77a^{28}+36a^{27}+18a^{26}+12a^{25}+5a^{24}+2a^{23}+a^{22}-a^{21}+a^{19}+2a^{18}+a^{17}-3a^{16}+a^{14}-3a^{13}+4a^{12}+3a^{11}-7a^{10}+3a^{9}+3a^{8}-9a^{7}+6a^{6}+5a^{5}-10a^{4}+7a^{3}+3a^{2}-13a+1215$, $7a^{30}-a^{29}-3a^{28}+8a^{27}-7a^{26}+2a^{25}+4a^{24}-6a^{23}+2a^{22}+5a^{21}-10a^{20}+9a^{19}-3a^{18}-3a^{17}+4a^{16}+a^{15}-7a^{14}+8a^{13}-2a^{12}-7a^{11}+11a^{10}-6a^{9}-5a^{8}+12a^{7}-7a^{6}-9a^{5}+25a^{4}-27a^{3}+11a^{2}+13a+1$, $6a^{30}+6a^{29}+4a^{28}+5a^{27}+4a^{26}+5a^{25}+4a^{24}+2a^{23}+4a^{22}+2a^{21}+4a^{20}+a^{19}+a^{18}+a^{17}-4a^{14}-a^{13}-3a^{12}-a^{11}-4a^{10}-6a^{9}-5a^{8}-5a^{7}-7a^{6}-7a^{5}-9a^{4}-6a^{3}-6a^{2}-10a+15$, $3a^{30}+4a^{29}+2a^{28}-5a^{26}-7a^{25}-2a^{24}+2a^{23}+3a^{22}+4a^{21}-6a^{19}-a^{18}+6a^{17}+5a^{16}+6a^{15}+3a^{14}-9a^{13}-10a^{12}-2a^{11}+2a^{10}+7a^{9}+8a^{8}-4a^{7}-11a^{6}-3a^{5}+a^{4}+3a^{3}+13a^{2}+3a-5$, $11a^{30}+3a^{29}+a^{28}+2a^{27}-3a^{26}+2a^{25}-4a^{24}+6a^{23}-3a^{22}+4a^{21}-3a^{20}+2a^{19}-3a^{17}+a^{16}-2a^{15}+3a^{14}-4a^{13}+6a^{12}-5a^{11}+11a^{10}-14a^{9}+8a^{8}-6a^{7}+9a^{5}-15a^{4}+26a^{3}-25a^{2}+16a+25$, $80a^{30}+203a^{29}+81a^{28}-154a^{27}-217a^{26}-5a^{25}+227a^{24}+196a^{23}-91a^{22}-289a^{21}-133a^{20}+201a^{19}+318a^{18}+33a^{17}-315a^{16}-294a^{15}+97a^{14}+409a^{13}+217a^{12}-256a^{11}-453a^{10}-91a^{9}+428a^{8}+435a^{7}-95a^{6}-570a^{5}-352a^{4}+335a^{3}+648a^{2}+184a-259$, $108a^{30}+60a^{29}+20a^{28}+18a^{27}+2a^{26}+10a^{25}-a^{24}+a^{23}-a^{22}-4a^{21}+9a^{20}-7a^{19}+10a^{18}-13a^{17}+9a^{16}-7a^{15}+10a^{14}-3a^{13}-3a^{12}+4a^{11}-10a^{10}+20a^{9}-16a^{8}+17a^{7}-22a^{6}+16a^{5}-8a^{4}+12a^{3}-15a+449$, $24a^{30}-14a^{28}+30a^{27}-14a^{26}-10a^{25}+17a^{24}+8a^{23}-38a^{22}+34a^{21}-25a^{19}+9a^{18}+32a^{17}-49a^{16}+21a^{15}+23a^{14}-36a^{13}-4a^{12}+52a^{11}-49a^{10}-11a^{9}+65a^{8}-43a^{7}-33a^{6}+67a^{5}-12a^{4}-78a^{3}+98a^{2}-23a+39$, $60a^{30}+26a^{29}-39a^{28}-71a^{27}-48a^{26}+27a^{25}+78a^{24}+62a^{23}-8a^{22}-75a^{21}-79a^{20}-12a^{19}+64a^{18}+99a^{17}+48a^{16}-53a^{15}-115a^{14}-91a^{13}+29a^{12}+128a^{11}+119a^{10}+3a^{9}-123a^{8}-147a^{7}-30a^{6}+95a^{5}+169a^{4}+82a^{3}-69a^{2}-185a+87$
| 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: | \( 26578047987565236000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 26578047987565236000 \cdot 1}{2\cdot\sqrt{949505273526965390612182983106676351835033630345588173537542144}}\cr\approx \mathstrut & 0.809976494830793 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 + 4*x - 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 + 4*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 + 4*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 + 4*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 $S_{31}$ |
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 | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $31$ | R | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $31$ | $20{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $16{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $31$ | $31$ | $1$ | $30$ | |||
|
\(7\)
| 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 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(195469\)
| $\Q_{195469}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{195469}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(212031575390627\)
| $\Q_{212031575390627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(304\!\cdots\!391\)
| $\Q_{30\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{30\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{30\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |