Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 1)
gp: K = bnfinit(y^20 - 12*y^18 + 57*y^16 - 136*y^14 + 168*y^12 - 90*y^10 - 14*y^8 + 51*y^6 - 35*y^4 + 10*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 1)
\( x^{20} - 12x^{18} + 57x^{16} - 136x^{14} + 168x^{12} - 90x^{10} - 14x^{8} + 51x^{6} - 35x^{4} + 10x^{2} - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[14, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-5278844430406469058634448896\)
\(\medspace = -\,2^{20}\cdot 11^{16}\cdot 331^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.33\) | 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: |
\(2\), \(11\), \(331\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
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: | $16$ | 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: |
$a$, $a^{19}-12a^{17}+57a^{15}-136a^{13}+168a^{11}-90a^{9}-14a^{7}+51a^{5}-35a^{3}+9a$, $16a^{18}-187a^{16}+855a^{14}-1923a^{12}+2138a^{10}-851a^{8}-446a^{6}+687a^{4}-369a^{2}+60$, $8a^{18}-95a^{16}+443a^{14}-1021a^{12}+1172a^{10}-498a^{8}-225a^{6}+377a^{4}-210a^{2}+36$, $a^{18}-11a^{16}+46a^{14}-90a^{12}+78a^{10}-12a^{8}-26a^{6}+25a^{4}-11a^{2}+1$, $17a^{19}-200a^{17}+921a^{15}-2086a^{13}+2332a^{11}-931a^{9}-482a^{7}+744a^{5}-406a^{3}+66a$, $17a^{18}-198a^{16}+901a^{14}-2013a^{12}+2216a^{10}-863a^{8}-472a^{6}+712a^{4}-380a^{2}+61$, $16a^{19}-189a^{17}+875a^{15}-1996a^{13}+2254a^{11}-919a^{9}-456a^{7}+718a^{5}-392a^{3}+65a$, $a+1$, $a^{19}+8a^{18}-12a^{17}-95a^{16}+57a^{15}+443a^{14}-136a^{13}-1021a^{12}+168a^{11}+1172a^{10}-90a^{9}-498a^{8}-14a^{7}-225a^{6}+51a^{5}+377a^{4}-35a^{3}-210a^{2}+10a+36$, $a^{2}+a-1$, $29a^{19}-13a^{18}-339a^{17}+152a^{16}+1549a^{15}-695a^{14}-3476a^{13}+1562a^{12}+3843a^{11}-1733a^{10}-1504a^{9}+687a^{8}-815a^{7}+359a^{6}+1234a^{5}-555a^{4}-661a^{3}+302a^{2}+103a-48$, $a^{19}+16a^{18}-12a^{17}-187a^{16}+57a^{15}+855a^{14}-136a^{13}-1923a^{12}+168a^{11}+2138a^{10}-90a^{9}-851a^{8}-14a^{7}-446a^{6}+51a^{5}+687a^{4}-35a^{3}-369a^{2}+9a+60$, $4a^{18}-48a^{16}+226a^{14}-524a^{12}+599a^{10}-244a^{8}-123a^{6}+189a^{4}-103a^{2}-a+16$, $4a^{19}-16a^{18}-47a^{17}+186a^{16}+216a^{15}-844a^{14}-487a^{13}+1878a^{12}+537a^{11}-2056a^{10}-200a^{9}+794a^{8}-125a^{7}+439a^{6}+171a^{5}-661a^{4}-89a^{3}+352a^{2}+12a-56$, $32a^{19}-9a^{18}-377a^{17}+106a^{16}+1740a^{15}-489a^{14}-3956a^{13}+1111a^{12}+4454a^{11}-1250a^{10}-1814a^{9}+510a^{8}-901a^{7}+251a^{6}+1428a^{5}-402a^{4}-781a^{3}+221a^{2}+129a-37$
| 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: | \( 5173145.55109 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{3}\cdot 5173145.55109 \cdot 1}{2\cdot\sqrt{5278844430406469058634448896}}\cr\approx \mathstrut & 0.144682103709 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 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^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 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^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 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^20 - 12*x^18 + 57*x^16 - 136*x^14 + 168*x^12 - 90*x^10 - 14*x^8 + 51*x^6 - 35*x^4 + 10*x^2 - 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
$C_2^{10}.C_2\wr C_5$ (as 20T846):
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 solvable group of order 163840 |
| The 649 conjugacy class representatives for $C_2^{10}.C_2\wr C_5$ |
| Character table for $C_2^{10}.C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.8.70952789611.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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]
Sibling fields
| Degree 20 siblings: | data not computed |
| Minimal sibling: | 20.6.5278844430406469058634448896.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | $20$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | $20$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(11\)
| 11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |
|
\(331\)
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |