Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*x^2 - 1)
gp: K = bnfinit(y^20 + 7*y^18 + 8*y^16 - 34*y^14 - 108*y^12 - 112*y^10 - 13*y^8 + 43*y^6 + 14*y^4 - 4*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*x^2 - 1)
\( x^{20} + 7x^{18} + 8x^{16} - 34x^{14} - 108x^{12} - 112x^{10} - 13x^{8} + 43x^{6} + 14x^{4} - 4x^{2} - 1 \)
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: | $[6, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-24868623129665465017517056\)
\(\medspace = -\,2^{10}\cdot 11^{16}\cdot 727^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.61\) | 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: | $2^{31/16}11^{4/5}727^{1/2}\approx 703.2778998665901$ | ||
| Ramified primes: |
\(2\), \(11\), \(727\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{243982}a^{18}+\frac{5909}{121991}a^{16}-\frac{97289}{243982}a^{14}+\frac{74807}{243982}a^{12}-\frac{1}{2}a^{11}-\frac{17722}{121991}a^{10}-\frac{1}{2}a^{9}+\frac{21958}{121991}a^{8}-\frac{1}{2}a^{7}-\frac{13869}{243982}a^{6}-\frac{47397}{121991}a^{4}-\frac{1}{2}a^{3}+\frac{10739}{121991}a^{2}-\frac{1}{2}a+\frac{57365}{243982}$, $\frac{1}{243982}a^{19}+\frac{5909}{121991}a^{17}+\frac{12351}{121991}a^{15}+\frac{74807}{243982}a^{13}+\frac{86547}{243982}a^{11}-\frac{1}{2}a^{10}+\frac{21958}{121991}a^{9}-\frac{1}{2}a^{8}-\frac{13869}{243982}a^{7}-\frac{1}{2}a^{6}-\frac{47397}{121991}a^{5}+\frac{10739}{121991}a^{3}-\frac{1}{2}a^{2}-\frac{32313}{121991}a-\frac{1}{2}$
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: | $12$ | 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: |
$a$, $\frac{96989}{121991}a^{19}+\frac{598521}{121991}a^{17}+\frac{285011}{121991}a^{15}-\frac{3496341}{121991}a^{13}-\frac{7535178}{121991}a^{11}-\frac{4826481}{121991}a^{9}+\frac{2128163}{121991}a^{7}+\frac{1950296}{121991}a^{5}+\frac{11426}{121991}a^{3}+\frac{130448}{121991}a$, $\frac{356907}{121991}a^{19}+\frac{2283939}{121991}a^{17}+\frac{1491036}{121991}a^{15}-\frac{12983339}{121991}a^{13}-\frac{30730722}{121991}a^{11}-\frac{21800221}{121991}a^{9}+\frac{7893048}{121991}a^{7}+\frac{10345044}{121991}a^{5}-\frac{997840}{121991}a^{3}-\frac{877394}{121991}a$, $\frac{134008}{121991}a^{18}+\frac{873319}{121991}a^{16}+\frac{649786}{121991}a^{14}-\frac{4875600}{121991}a^{12}-\frac{12137076}{121991}a^{10}-\frac{9143819}{121991}a^{8}+\frac{2781726}{121991}a^{6}+\frac{4648118}{121991}a^{4}-\frac{31830}{121991}a^{2}-\frac{381909}{121991}$, $\frac{25962}{121991}a^{18}+\frac{133542}{121991}a^{16}-\frac{115354}{121991}a^{14}-\frac{1177296}{121991}a^{12}-\frac{1116934}{121991}a^{10}+\frac{1727180}{121991}a^{8}+\frac{3832175}{121991}a^{6}+\frac{1224516}{121991}a^{4}-\frac{740971}{121991}a^{2}-\frac{199980}{121991}$, $\frac{37019}{243982}a^{19}-\frac{85271}{121991}a^{18}+\frac{137399}{121991}a^{17}-\frac{574982}{121991}a^{16}+\frac{364775}{243982}a^{15}-\frac{555600}{121991}a^{14}-\frac{1379259}{243982}a^{13}+\frac{5816963}{243982}a^{12}-\frac{2300949}{121991}a^{11}+\frac{16749365}{243982}a^{10}-\frac{2158669}{121991}a^{9}+\frac{16217785}{243982}a^{8}+\frac{653563}{243982}a^{7}+\frac{774691}{121991}a^{6}+\frac{1348911}{121991}a^{5}-\frac{4646621}{243982}a^{4}-\frac{21628}{121991}a^{3}-\frac{1097229}{243982}a^{2}-\frac{512357}{243982}a+\frac{146194}{121991}$, $\frac{37019}{243982}a^{19}+\frac{85271}{121991}a^{18}+\frac{137399}{121991}a^{17}+\frac{574982}{121991}a^{16}+\frac{364775}{243982}a^{15}+\frac{555600}{121991}a^{14}-\frac{1379259}{243982}a^{13}-\frac{5816963}{243982}a^{12}-\frac{2300949}{121991}a^{11}-\frac{16749365}{243982}a^{10}-\frac{2158669}{121991}a^{9}-\frac{16217785}{243982}a^{8}+\frac{653563}{243982}a^{7}-\frac{774691}{121991}a^{6}+\frac{1348911}{121991}a^{5}+\frac{4646621}{243982}a^{4}-\frac{21628}{121991}a^{3}+\frac{1097229}{243982}a^{2}-\frac{512357}{243982}a-\frac{146194}{121991}$, $\frac{71111}{243982}a^{19}+\frac{34451}{243982}a^{18}+\frac{179886}{121991}a^{17}+\frac{89971}{121991}a^{16}-\frac{165230}{121991}a^{15}-\frac{122605}{243982}a^{14}-\frac{3110735}{243982}a^{13}-\frac{1469801}{243982}a^{12}-\frac{1404013}{121991}a^{11}-\frac{1537217}{243982}a^{10}+\frac{4454743}{243982}a^{9}+\frac{1603617}{243982}a^{8}+\frac{4665046}{121991}a^{7}+\frac{1970770}{121991}a^{6}+\frac{1017200}{121991}a^{5}+\frac{463452}{121991}a^{4}-\frac{2323091}{243982}a^{3}-\frac{1156747}{243982}a^{2}-\frac{340507}{243982}a-\frac{47288}{121991}$, $\frac{176916}{121991}a^{18}+\frac{1087458}{121991}a^{16}+\frac{461412}{121991}a^{14}-\frac{6591910}{121991}a^{12}-\frac{13692314}{121991}a^{10}-\frac{7239212}{121991}a^{8}+\frac{6666484}{121991}a^{6}+\frac{4773079}{121991}a^{4}-\frac{1071739}{121991}a^{2}-\frac{376896}{121991}$, $\frac{302467}{243982}a^{19}+\frac{133523}{243982}a^{18}+\frac{963290}{121991}a^{17}+\frac{437583}{121991}a^{16}+\frac{1176967}{243982}a^{15}+\frac{746425}{243982}a^{14}-\frac{11221001}{243982}a^{13}-\frac{2218852}{121991}a^{12}-\frac{25811369}{243982}a^{11}-\frac{12147467}{243982}a^{10}-\frac{16684021}{243982}a^{9}-\frac{11900447}{243982}a^{8}+\frac{4752496}{121991}a^{7}-\frac{2202945}{243982}a^{6}+\frac{5111570}{121991}a^{5}+\frac{1948799}{243982}a^{4}-\frac{1220407}{243982}a^{3}+\frac{1140485}{243982}a^{2}-\frac{429197}{121991}a-\frac{24013}{243982}$, $\frac{26321}{243982}a^{19}+\frac{89336}{121991}a^{18}+\frac{106519}{243982}a^{17}+\frac{1223387}{243982}a^{16}-\frac{137335}{121991}a^{15}+\frac{1263265}{243982}a^{14}-\frac{1159603}{243982}a^{13}-\frac{6127161}{243982}a^{12}-\frac{56347}{243982}a^{11}-\frac{18230235}{243982}a^{10}+\frac{1671034}{121991}a^{9}-\frac{8978118}{121991}a^{8}+\frac{1988413}{121991}a^{7}-\frac{792334}{121991}a^{6}-\frac{478915}{243982}a^{5}+\frac{3026220}{121991}a^{4}-\frac{1569739}{243982}a^{3}+\frac{1510221}{243982}a^{2}-\frac{344415}{243982}a-\frac{286531}{243982}$, $\frac{230997}{243982}a^{19}+\frac{85271}{121991}a^{18}+\frac{735920}{121991}a^{17}+\frac{574982}{121991}a^{16}+\frac{934797}{243982}a^{15}+\frac{555600}{121991}a^{14}-\frac{8371941}{243982}a^{13}-\frac{5816963}{243982}a^{12}-\frac{9836127}{121991}a^{11}-\frac{16749365}{243982}a^{10}-\frac{6985150}{121991}a^{9}-\frac{16217785}{243982}a^{8}+\frac{4909889}{243982}a^{7}-\frac{774691}{121991}a^{6}+\frac{3299207}{121991}a^{5}+\frac{4646621}{243982}a^{4}-\frac{10202}{121991}a^{3}+\frac{1097229}{243982}a^{2}-\frac{251461}{243982}a-\frac{146194}{121991}$
| 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: | \( 61407.6347505 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{7}\cdot 61407.6347505 \cdot 1}{2\cdot\sqrt{24868623129665465017517056}}\cr\approx \mathstrut & 0.152337055275 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*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 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*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 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*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 + 7*x^18 + 8*x^16 - 34*x^14 - 108*x^12 - 112*x^10 - 13*x^8 + 43*x^6 + 14*x^4 - 4*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
| \(\Q(\zeta_{11})^+\), 10.8.155838906487.1 |
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: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.13 | $x^{10} + 10 x^{9} + 10 x^{8} + 56 x^{7} + 192 x^{6} + 800 x^{5} + 1536 x^{4} + 2208 x^{3} + 2224 x^{2} + 96 x - 1056$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
|
\(11\)
| 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 | $[\ ]_{5}^{4}$ |
|
\(727\)
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |