Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 2*x - 5)
gp: K = bnfinit(y^31 + 2*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + 2*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 2*x - 5)
\( x^{31} + 2x - 5 \)
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: |
\(-15896907197985528699344249971836779878127944915555417537689208984375\)
\(\medspace = -\,5^{30}\cdot 11\cdot 71\cdot 311\cdot 1667\cdot 209394034439\cdot 201326610321383981030559113\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(147.16\) | 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: | not computed | ||
| Ramified primes: |
\(5\), \(11\), \(71\), \(311\), \(1667\), \(209394034439\), \(201326610321383981030559113\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-17069\!\cdots\!97279}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{30}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+3a-4$, $24a^{30}+21a^{29}+17a^{28}+2a^{27}-5a^{26}-13a^{25}-21a^{24}-28a^{23}-20a^{22}+a^{21}+23a^{20}+41a^{19}+27a^{18}+14a^{17}-16a^{15}-22a^{14}-41a^{13}-39a^{12}-24a^{11}+31a^{10}+39a^{9}+67a^{8}+24a^{7}+11a^{6}-11a^{5}-23a^{4}-53a^{3}-65a^{2}-50a+38$, $53a^{30}+3a^{29}-36a^{28}+2a^{27}+54a^{26}+28a^{25}-45a^{24}-51a^{23}+23a^{22}+50a^{21}-31a^{20}-97a^{19}-34a^{18}+61a^{17}+19a^{16}-108a^{15}-101a^{14}+45a^{13}+90a^{12}-53a^{11}-143a^{10}+152a^{8}+54a^{7}-131a^{6}-73a^{5}+160a^{4}+184a^{3}-52a^{2}-146a+182$, $34a^{30}+72a^{29}+23a^{28}+145a^{27}+15a^{26}+101a^{25}+18a^{24}-16a^{23}+75a^{22}-69a^{21}+127a^{20}-37a^{19}+36a^{18}+a^{17}-165a^{16}+24a^{15}-245a^{14}+23a^{13}-112a^{12}-91a^{11}+45a^{10}-295a^{9}+60a^{8}-316a^{7}-18a^{6}+a^{5}-118a^{4}+346a^{3}-209a^{2}+326a-42$, $34a^{30}-3a^{29}+45a^{28}-10a^{27}+45a^{26}-13a^{25}+52a^{24}-14a^{23}+58a^{22}-12a^{21}+56a^{20}-18a^{19}+58a^{18}-2a^{17}+56a^{16}+3a^{15}+54a^{14}+6a^{13}+29a^{12}+32a^{11}+33a^{10}+40a^{9}+16a^{8}+74a^{7}-18a^{6}+89a^{5}-13a^{4}+123a^{3}-65a^{2}+157a-17$, $12a^{30}+73a^{29}-20a^{28}-77a^{27}+28a^{26}+85a^{25}-40a^{24}-92a^{23}+54a^{22}+98a^{21}-66a^{20}-108a^{19}+84a^{18}+116a^{17}-102a^{16}-119a^{15}+118a^{14}+125a^{13}-144a^{12}-125a^{11}+173a^{10}+120a^{9}-199a^{8}-120a^{7}+240a^{6}+113a^{5}-282a^{4}-102a^{3}+318a^{2}+103a-339$, $6a^{30}+4a^{29}-8a^{28}-16a^{27}-7a^{26}+7a^{25}+5a^{24}-10a^{23}-22a^{22}-13a^{21}+11a^{20}+19a^{19}+4a^{18}-17a^{17}-18a^{16}+9a^{15}+28a^{14}+16a^{13}-9a^{12}-14a^{11}+15a^{10}+39a^{9}+23a^{8}-15a^{7}-31a^{6}+a^{5}+42a^{4}+40a^{3}-7a^{2}-52a-21$, $6a^{30}+110a^{29}-87a^{28}+94a^{27}+45a^{26}-91a^{25}+208a^{24}-123a^{23}+59a^{22}+161a^{21}-224a^{20}+317a^{19}-138a^{18}-11a^{17}+310a^{16}-364a^{15}+399a^{14}-101a^{13}-121a^{12}+462a^{11}-454a^{10}+393a^{9}+29a^{8}-270a^{7}+567a^{6}-412a^{5}+221a^{4}+296a^{3}-448a^{2}+562a-136$, $65a^{30}+59a^{29}+78a^{28}+61a^{27}+95a^{26}+53a^{25}+98a^{24}+58a^{23}+93a^{22}+88a^{21}+91a^{20}+108a^{19}+77a^{18}+124a^{17}+71a^{16}+148a^{15}+87a^{14}+153a^{13}+89a^{12}+139a^{11}+114a^{10}+116a^{9}+185a^{8}+88a^{7}+213a^{6}+76a^{5}+214a^{4}+103a^{3}+231a^{2}+161a+323$, $807a^{30}+858a^{29}+833a^{28}+953a^{27}+843a^{26}+1047a^{25}+914a^{24}+1115a^{23}+991a^{22}+1140a^{21}+1140a^{20}+1169a^{19}+1285a^{18}+1155a^{17}+1451a^{16}+1214a^{15}+1583a^{14}+1276a^{13}+1638a^{12}+1478a^{11}+1664a^{10}+1700a^{9}+1599a^{8}+1981a^{7}+1635a^{6}+2217a^{5}+1655a^{4}+2358a^{3}+1884a^{2}+2441a+3777$, $18a^{30}-a^{29}-16a^{28}+12a^{27}-27a^{26}-32a^{25}-14a^{24}-53a^{23}-37a^{22}-15a^{21}-60a^{20}-7a^{19}-2a^{18}-48a^{17}+12a^{16}-2a^{15}-45a^{14}+32a^{13}-11a^{12}-12a^{11}+83a^{10}+8a^{9}+41a^{8}+128a^{7}+12a^{6}+57a^{5}+107a^{4}-39a^{3}+66a^{2}+47a-33$, $71a^{30}+162a^{29}+184a^{28}+151a^{27}+75a^{26}-59a^{25}-186a^{24}-228a^{23}-201a^{22}-119a^{21}+32a^{20}+190a^{19}+271a^{18}+260a^{17}+178a^{16}+25a^{15}-180a^{14}-326a^{13}-339a^{12}-266a^{11}-101a^{10}+174a^{9}+387a^{8}+420a^{7}+353a^{6}+182a^{5}-132a^{4}-421a^{3}-509a^{2}-458a-169$, $12a^{30}+7a^{29}+3a^{28}-2a^{27}-3a^{26}-6a^{25}-8a^{24}-9a^{23}-13a^{22}-15a^{21}-22a^{20}-25a^{19}-25a^{18}-17a^{17}+2a^{16}+20a^{15}+43a^{14}+49a^{13}+49a^{12}+31a^{11}+8a^{10}-19a^{9}-50a^{8}-67a^{7}-78a^{6}-70a^{5}-39a^{4}-12a^{3}+36a^{2}+47a+86$, $23a^{30}+39a^{29}+29a^{28}+44a^{27}+45a^{26}+33a^{25}+48a^{24}+33a^{23}+46a^{22}+56a^{21}+40a^{20}+58a^{19}+31a^{18}+31a^{17}+51a^{16}+29a^{15}+75a^{14}+62a^{13}+49a^{12}+87a^{11}+36a^{10}+78a^{9}+94a^{8}+50a^{7}+120a^{6}+58a^{5}+59a^{4}+110a^{3}+28a^{2}+106a+127$, $74a^{30}+44a^{29}-26a^{28}-70a^{27}-86a^{26}-38a^{25}+28a^{24}+94a^{23}+90a^{22}+43a^{21}-46a^{20}-106a^{19}-111a^{18}-38a^{17}+52a^{16}+119a^{15}+115a^{14}+30a^{13}-75a^{12}-144a^{11}-116a^{10}-20a^{9}+134a^{8}+190a^{7}+161a^{6}-6a^{5}-155a^{4}-282a^{3}-188a^{2}-39a+351$
| 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: | \( 4536078693224075000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 4536078693224075000000 \cdot 1}{2\cdot\sqrt{15896907197985528699344249971836779878127944915555417537689208984375}}\cr\approx \mathstrut & 1.06837148008319 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 + 2*x - 5)
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 + 2*x - 5, 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 + 2*x - 5);
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 + 2*x - 5);
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 | ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $18{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/47.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.10.10.2 | $x^{10} - 20 x^{6} + 10 x^{5} + 50 x^{2} - 100 x + 25$ | $5$ | $2$ | $10$ | $C_5^2 : C_8$ | $[5/4, 5/4]_{4}^{2}$ | |
| 5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
| 5.10.10.13 | $x^{10} - 10 x^{6} + 10 x^{5} - 25 x^{2} - 50 x + 25$ | $5$ | $2$ | $10$ | $(C_5^2 : C_4) : C_2$ | $[5/4, 5/4]_{4}^{2}$ | |
|
\(11\)
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.12.0.1 | $x^{12} + x^{8} + x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 5 x^{3} + 6 x^{2} + 5 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 11.14.0.1 | $x^{14} + 2 x^{7} + 9 x^{6} + 6 x^{5} + 4 x^{4} + 8 x^{3} + 6 x^{2} + 10 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 71.23.0.1 | $x^{23} + 4 x + 64$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(311\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(1667\)
| $\Q_{1667}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1667}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1667}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(209394034439\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(201\!\cdots\!113\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |