Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 2*x + 4)
gp: K = bnfinit(y^30 - 2*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 2*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 2*x + 4)
\( x^{30} - 2x + 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-14836019611796354892170677662045618282985013768744056322523136\)
\(\medspace = -\,2^{28}\cdot 4899563\cdot 468731357\cdot 5400051347\cdot 4456544800255361855961256703\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(109.41\) | 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: | $2^{28/29}4899563^{1/2}468731357^{1/2}5400051347^{1/2}4456544800255361855961256703^{1/2}\approx 4.590800402718109e+26$ | ||
| Ramified primes: |
\(2\), \(4899563\), \(468731357\), \(5400051347\), \(4456544800255361855961256703\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-55268\!\cdots\!16531}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{2}a^{29}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $14$ | 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: |
$9a^{29}+13a^{28}+11a^{27}+7a^{26}+3a^{25}-4a^{24}-12a^{23}-13a^{22}-13a^{21}-12a^{20}-2a^{19}+6a^{18}+10a^{17}+16a^{16}+19a^{15}+12a^{14}+4a^{13}-a^{12}-15a^{11}-22a^{10}-18a^{9}-18a^{8}-11a^{7}+5a^{6}+15a^{5}+18a^{4}+28a^{3}+27a^{2}+7a-17$, $2a^{29}-5a^{27}-8a^{26}-3a^{25}+5a^{24}+9a^{23}+8a^{22}+2a^{21}-3a^{20}-2a^{19}-2a^{18}-5a^{17}-8a^{16}-9a^{15}+13a^{13}+15a^{12}+7a^{11}-5a^{10}-13a^{9}-6a^{8}+6a^{7}+6a^{6}-7a^{4}-8a^{3}+4a^{2}+8a-5$, $13a^{29}+6a^{28}+3a^{27}-5a^{26}-8a^{25}-12a^{24}-20a^{23}-19a^{22}-27a^{21}-19a^{20}-19a^{19}-13a^{18}-6a^{17}-6a^{16}+9a^{15}+11a^{14}+23a^{13}+26a^{12}+22a^{11}+29a^{10}+18a^{9}+24a^{8}+13a^{7}-2a^{5}-27a^{4}-13a^{3}-35a^{2}-27a-61$, $10a^{29}+16a^{28}+2a^{27}+17a^{26}-3a^{25}+14a^{24}-3a^{23}+9a^{22}-2a^{21}-4a^{20}-19a^{18}+3a^{17}-28a^{16}+2a^{15}-32a^{14}-2a^{13}-29a^{12}-20a^{11}-22a^{10}-40a^{9}-11a^{8}-53a^{7}-61a^{5}+2a^{4}-52a^{3}-16a^{2}-37a-59$, $25a^{29}+11a^{28}+4a^{27}+15a^{26}+30a^{25}+29a^{24}+6a^{23}-19a^{22}-25a^{21}-10a^{20}+2a^{19}-9a^{18}-37a^{17}-51a^{16}-32a^{15}+4a^{14}+21a^{13}+5a^{12}-17a^{11}-9a^{10}+33a^{9}+69a^{8}+64a^{7}+22a^{6}-8a^{5}+5a^{4}+46a^{3}+54a^{2}+7a-111$, $8a^{29}+8a^{28}+2a^{27}+5a^{26}+7a^{25}+8a^{24}+4a^{23}-a^{22}+4a^{21}+6a^{20}+5a^{19}-2a^{18}-2a^{17}+6a^{15}-3a^{14}-7a^{13}-7a^{12}-3a^{11}+3a^{10}-13a^{9}-11a^{8}-12a^{7}-a^{6}-6a^{5}-15a^{4}-18a^{3}-10a^{2}-2a-29$, $7a^{29}+17a^{28}+9a^{27}-22a^{26}+12a^{25}-3a^{24}+13a^{23}-20a^{22}+18a^{21}+5a^{20}-15a^{19}-15a^{18}+45a^{16}-34a^{15}-12a^{14}-27a^{13}+56a^{12}-10a^{11}-19a^{10}-6a^{9}+8a^{8}+24a^{7}-50a^{6}+64a^{5}-6a^{4}+25a^{3}-97a^{2}+41a+47$, $a^{29}+7a^{28}+4a^{27}-5a^{26}-a^{25}-7a^{24}+8a^{23}-5a^{22}+12a^{21}-15a^{20}+16a^{19}-16a^{18}+19a^{17}-24a^{16}+19a^{15}-27a^{14}+37a^{13}-22a^{12}+30a^{11}-43a^{10}+19a^{9}-33a^{8}+51a^{7}-19a^{6}+31a^{5}-47a^{4}+6a^{3}-22a^{2}+40a-1$, $15a^{29}-25a^{28}+26a^{27}-29a^{26}+27a^{25}-28a^{24}+23a^{23}-11a^{22}-5a^{21}+21a^{20}-32a^{19}+47a^{18}-54a^{17}+60a^{16}-48a^{15}+41a^{14}-24a^{13}+18a^{12}+9a^{11}-29a^{10}+62a^{9}-74a^{8}+99a^{7}-103a^{6}+112a^{5}-82a^{4}+51a^{3}-29a+45$, $9a^{29}+4a^{28}-9a^{27}-4a^{26}+16a^{25}+21a^{24}-7a^{23}-12a^{22}+19a^{21}+10a^{20}-26a^{19}-19a^{18}+8a^{17}+11a^{16}-13a^{15}-22a^{14}+25a^{13}+40a^{12}-18a^{11}-22a^{10}+29a^{9}+18a^{8}-30a^{7}-41a^{6}+2a^{5}+36a^{4}-27a^{3}-54a^{2}+45a+45$, $20a^{29}+37a^{28}-45a^{27}+25a^{26}+37a^{25}-92a^{24}+57a^{23}+8a^{22}-64a^{21}+84a^{20}-2a^{19}-90a^{18}+98a^{17}-51a^{16}-65a^{15}+149a^{14}-82a^{13}-31a^{12}+123a^{11}-132a^{10}-29a^{9}+184a^{8}-195a^{7}+101a^{6}+99a^{5}-209a^{4}+69a^{3}+142a^{2}-304a+253$, $15a^{29}+5a^{28}-16a^{27}-8a^{26}+15a^{25}+11a^{24}-14a^{23}-15a^{22}+14a^{21}+17a^{20}-16a^{19}-15a^{18}+17a^{17}+10a^{16}-15a^{15}-6a^{14}+12a^{13}+8a^{12}-13a^{11}-13a^{10}+22a^{9}+18a^{8}-34a^{7}-19a^{6}+45a^{5}+21a^{4}-48a^{3}-28a^{2}+50a+11$, $45a^{29}-56a^{28}+71a^{27}-72a^{26}+74a^{25}-60a^{24}+40a^{23}-11a^{22}-26a^{21}+59a^{20}-92a^{19}+114a^{18}-125a^{17}+130a^{16}-116a^{15}+100a^{14}-62a^{13}+19a^{12}+38a^{11}-103a^{10}+161a^{9}-219a^{8}+245a^{7}-253a^{6}+223a^{5}-165a^{4}+90a^{3}+6a^{2}-99a+109$, $23a^{29}+5a^{28}-20a^{27}+2a^{26}+17a^{25}-11a^{24}-13a^{23}+16a^{22}-3a^{21}-35a^{20}+4a^{19}+43a^{18}+7a^{17}-12a^{16}+20a^{15}-8a^{14}-66a^{13}-16a^{12}+63a^{11}+31a^{10}-5a^{9}+38a^{8}+10a^{7}-87a^{6}-48a^{5}+61a^{4}+28a^{3}-23a^{2}+56a+9$
| 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: | \( 16701494058586538000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 16701494058586538000 \cdot 1}{2\cdot\sqrt{14836019611796354892170677662045618282985013768744056322523136}}\cr\approx \mathstrut & 2.03593849378485 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 2*x + 4)
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^30 - 2*x + 4, 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^30 - 2*x + 4);
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^30 - 2*x + 4);
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 265252859812191058636308480000000 |
| The 5604 conjugacy class representatives for $S_{30}$ are not computed |
| Character table for $S_{30}$ 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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{10}$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | $25{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $20{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $29$ | $29$ | $1$ | $28$ | ||||
|
\(4899563\)
| $\Q_{4899563}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{4899563}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(468731357\)
| $\Q_{468731357}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(5400051347\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(445\!\cdots\!703\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |