Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*x^2 + 1)
gp: K = bnfinit(y^20 - 12*y^18 + 54*y^16 - 105*y^14 + 50*y^12 + 110*y^10 - 146*y^8 + 37*y^6 + 18*y^4 - 9*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 12*x^18 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 12*x^18 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*x^2 + 1)
\( x^{20} - 12x^{18} + 54x^{16} - 105x^{14} + 50x^{12} + 110x^{10} - 146x^{8} + 37x^{6} + 18x^{4} - 9x^{2} + 1 \)
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: | $[16, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(437402377341742634765604880384\)
\(\medspace = 2^{20}\cdot 11^{16}\cdot 23^{2}\cdot 131^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.34\) | 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: |
\(2\), \(11\), \(23\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: | $17$ | 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: |
$a^{19}-12a^{17}+54a^{15}-105a^{13}+50a^{11}+110a^{9}-146a^{7}+37a^{5}+18a^{3}-9a$, $3a^{19}-35a^{17}+150a^{15}-262a^{13}+54a^{11}+354a^{9}-310a^{7}-5a^{5}+54a^{3}-8a$, $2a^{18}-24a^{16}+107a^{14}-201a^{12}+73a^{10}+244a^{8}-270a^{6}+30a^{4}+49a^{2}-11$, $2a^{18}-24a^{16}+108a^{14}-209a^{12}+92a^{10}+239a^{8}-297a^{6}+47a^{4}+52a^{2}-12$, $5a^{18}-59a^{16}+258a^{14}-471a^{12}+146a^{10}+593a^{8}-607a^{6}+42a^{4}+106a^{2}-21$, $7a^{19}-82a^{17}+354a^{15}-629a^{13}+157a^{11}+824a^{9}-774a^{7}+21a^{5}+135a^{3}-23a$, $5a^{19}-58a^{17}+247a^{15}-428a^{13}+84a^{11}+580a^{9}-504a^{7}-9a^{5}+86a^{3}-12a$, $4a^{18}-47a^{16}+204a^{14}-367a^{12}+103a^{10}+470a^{8}-464a^{6}+26a^{4}+82a^{2}-17$, $2a^{18}-24a^{16}+107a^{14}-201a^{12}+73a^{10}+244a^{8}-270a^{6}+31a^{4}+46a^{2}-11$, $a+1$, $a^{2}-a-1$, $5a^{18}-59a^{16}+258a^{14}-471a^{12}+146a^{10}+593a^{8}-607a^{6}+42a^{4}+106a^{2}+a-21$, $3a^{19}+2a^{18}-36a^{17}-24a^{16}+162a^{15}+108a^{14}-314a^{13}-209a^{12}+142a^{11}+92a^{10}+349a^{9}+239a^{8}-442a^{7}-297a^{6}+80a^{5}+46a^{4}+72a^{3}+55a^{2}-18a-13$, $5a^{19}-6a^{18}-58a^{17}+71a^{16}+247a^{15}-311a^{14}-428a^{13}+568a^{12}+84a^{11}-177a^{10}+580a^{9}-708a^{8}-504a^{7}+726a^{6}-9a^{5}-62a^{4}+86a^{3}-121a^{2}-12a+28$, $8a^{18}-94a^{16}+408a^{14}-734a^{12}+207a^{10}+934a^{8}-920a^{6}+57a^{4}+156a^{2}-a-32$, $11a^{19}-5a^{18}-130a^{17}+59a^{16}+570a^{15}-258a^{14}-1048a^{13}+472a^{12}+349a^{11}-153a^{10}+1283a^{9}-580a^{8}-1363a^{7}+610a^{6}+141a^{5}-62a^{4}+227a^{3}-102a^{2}-55a+24$, $9a^{19}-13a^{18}-106a^{17}+153a^{16}+462a^{15}-666a^{14}-838a^{13}+1205a^{12}+249a^{11}-353a^{10}+1063a^{9}-1527a^{8}-1070a^{7}+1527a^{6}+64a^{5}-99a^{4}+189a^{3}-262a^{2}-33a+52$
| 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: | \( 78852501.6337 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{2}\cdot 78852501.6337 \cdot 1}{2\cdot\sqrt{437402377341742634765604880384}}\cr\approx \mathstrut & 0.154235582565 \end{aligned}\] (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 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*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 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*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 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*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 + 54*x^16 - 105*x^14 + 50*x^12 + 110*x^10 - 146*x^8 + 37*x^6 + 18*x^4 - 9*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
$C_2^5.C_2^8:C_{10}$ (as 20T751):
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 81920 |
| The 332 conjugacy class representatives for $C_2^5.C_2^8:C_{10}$ |
| Character table for $C_2^5.C_2^8:C_{10}$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.10.645863308453.1 |
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: | 20.4.148655836175885980086285893632.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |