Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279)
gp: K = bnfinit(y^10 - 2*y^9 - 122*y^8 + 198*y^7 + 5582*y^6 - 6794*y^5 - 118785*y^4 + 94556*y^3 + 1168426*y^2 - 446344*y - 4240279, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279)
\( x^{10} - 2 x^{9} - 122 x^{8} + 198 x^{7} + 5582 x^{6} - 6794 x^{5} - 118785 x^{4} + 94556 x^{3} + \cdots - 4240279 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1412799778009275392\)
\(\medspace = 2^{10}\cdot 11^{8}\cdot 23^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.31\) | 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: | $2\cdot 11^{4/5}23^{1/2}\approx 65.31426768087508$ | ||
| Ramified primes: |
\(2\), \(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{23}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1012=2^{2}\cdot 11\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1012}(1,·)$, $\chi_{1012}(643,·)$, $\chi_{1012}(551,·)$, $\chi_{1012}(553,·)$, $\chi_{1012}(93,·)$, $\chi_{1012}(367,·)$, $\chi_{1012}(185,·)$, $\chi_{1012}(91,·)$, $\chi_{1012}(829,·)$, $\chi_{1012}(735,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{26\!\cdots\!37}a^{9}+\frac{13\!\cdots\!91}{26\!\cdots\!37}a^{8}-\frac{10\!\cdots\!05}{26\!\cdots\!37}a^{7}-\frac{43\!\cdots\!01}{26\!\cdots\!37}a^{6}+\frac{45\!\cdots\!07}{26\!\cdots\!37}a^{5}+\frac{12\!\cdots\!65}{26\!\cdots\!37}a^{4}-\frac{32\!\cdots\!20}{26\!\cdots\!37}a^{3}+\frac{12\!\cdots\!93}{26\!\cdots\!37}a^{2}-\frac{22\!\cdots\!49}{26\!\cdots\!37}a-\frac{34\!\cdots\!55}{26\!\cdots\!37}$
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: | $9$ | 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: |
$\frac{13486765050398}{26\!\cdots\!37}a^{9}-\frac{218606664356652}{26\!\cdots\!37}a^{8}-\frac{13\!\cdots\!18}{26\!\cdots\!37}a^{7}+\frac{28\!\cdots\!20}{26\!\cdots\!37}a^{6}+\frac{48\!\cdots\!12}{26\!\cdots\!37}a^{5}-\frac{15\!\cdots\!72}{26\!\cdots\!37}a^{4}-\frac{70\!\cdots\!10}{26\!\cdots\!37}a^{3}+\frac{36\!\cdots\!05}{26\!\cdots\!37}a^{2}+\frac{34\!\cdots\!04}{26\!\cdots\!37}a-\frac{25\!\cdots\!81}{26\!\cdots\!37}$, $\frac{338643523778168}{26\!\cdots\!37}a^{9}-\frac{40\!\cdots\!16}{26\!\cdots\!37}a^{8}-\frac{34\!\cdots\!28}{26\!\cdots\!37}a^{7}+\frac{39\!\cdots\!04}{26\!\cdots\!37}a^{6}+\frac{12\!\cdots\!16}{26\!\cdots\!37}a^{5}-\frac{13\!\cdots\!73}{26\!\cdots\!37}a^{4}-\frac{18\!\cdots\!36}{26\!\cdots\!37}a^{3}+\frac{17\!\cdots\!34}{26\!\cdots\!37}a^{2}+\frac{89\!\cdots\!24}{26\!\cdots\!37}a-\frac{77\!\cdots\!59}{26\!\cdots\!37}$, $\frac{430879143488320}{26\!\cdots\!37}a^{9}-\frac{699364575945516}{26\!\cdots\!37}a^{8}-\frac{51\!\cdots\!48}{26\!\cdots\!37}a^{7}+\frac{62\!\cdots\!01}{26\!\cdots\!37}a^{6}+\frac{21\!\cdots\!22}{26\!\cdots\!37}a^{5}-\frac{15\!\cdots\!60}{26\!\cdots\!37}a^{4}-\frac{35\!\cdots\!22}{26\!\cdots\!37}a^{3}+\frac{64\!\cdots\!86}{26\!\cdots\!37}a^{2}+\frac{20\!\cdots\!22}{26\!\cdots\!37}a+\frac{79\!\cdots\!35}{26\!\cdots\!37}$, $\frac{34848362377262}{26\!\cdots\!37}a^{9}-\frac{62953342229325}{26\!\cdots\!37}a^{8}-\frac{43\!\cdots\!90}{26\!\cdots\!37}a^{7}+\frac{62\!\cdots\!17}{26\!\cdots\!37}a^{6}+\frac{20\!\cdots\!94}{26\!\cdots\!37}a^{5}-\frac{21\!\cdots\!22}{26\!\cdots\!37}a^{4}-\frac{49\!\cdots\!56}{26\!\cdots\!37}a^{3}+\frac{29\!\cdots\!67}{26\!\cdots\!37}a^{2}+\frac{46\!\cdots\!46}{26\!\cdots\!37}a-\frac{13\!\cdots\!26}{26\!\cdots\!37}$, $\frac{155437192739182}{39\!\cdots\!11}a^{9}-\frac{17\!\cdots\!40}{39\!\cdots\!11}a^{8}-\frac{17\!\cdots\!74}{39\!\cdots\!11}a^{7}+\frac{18\!\cdots\!40}{39\!\cdots\!11}a^{6}+\frac{74\!\cdots\!66}{39\!\cdots\!11}a^{5}-\frac{63\!\cdots\!28}{39\!\cdots\!11}a^{4}-\frac{12\!\cdots\!72}{39\!\cdots\!11}a^{3}+\frac{92\!\cdots\!54}{39\!\cdots\!11}a^{2}+\frac{74\!\cdots\!62}{39\!\cdots\!11}a-\frac{46\!\cdots\!99}{39\!\cdots\!11}$, $\frac{91\!\cdots\!34}{26\!\cdots\!37}a^{9}+\frac{98\!\cdots\!46}{26\!\cdots\!37}a^{8}-\frac{11\!\cdots\!60}{26\!\cdots\!37}a^{7}-\frac{96\!\cdots\!48}{26\!\cdots\!37}a^{6}+\frac{46\!\cdots\!22}{26\!\cdots\!37}a^{5}+\frac{32\!\cdots\!12}{26\!\cdots\!37}a^{4}-\frac{69\!\cdots\!70}{26\!\cdots\!37}a^{3}-\frac{41\!\cdots\!92}{26\!\cdots\!37}a^{2}+\frac{33\!\cdots\!08}{26\!\cdots\!37}a+\frac{17\!\cdots\!55}{26\!\cdots\!37}$, $\frac{27\!\cdots\!90}{26\!\cdots\!37}a^{9}-\frac{91\!\cdots\!52}{26\!\cdots\!37}a^{8}-\frac{28\!\cdots\!92}{26\!\cdots\!37}a^{7}+\frac{79\!\cdots\!54}{26\!\cdots\!37}a^{6}+\frac{10\!\cdots\!98}{26\!\cdots\!37}a^{5}-\frac{21\!\cdots\!72}{26\!\cdots\!37}a^{4}-\frac{15\!\cdots\!08}{26\!\cdots\!37}a^{3}+\frac{17\!\cdots\!46}{26\!\cdots\!37}a^{2}+\frac{81\!\cdots\!44}{26\!\cdots\!37}a-\frac{71\!\cdots\!69}{26\!\cdots\!37}$, $\frac{78\!\cdots\!86}{26\!\cdots\!37}a^{9}+\frac{48\!\cdots\!56}{26\!\cdots\!37}a^{8}-\frac{79\!\cdots\!06}{26\!\cdots\!37}a^{7}-\frac{44\!\cdots\!82}{26\!\cdots\!37}a^{6}+\frac{26\!\cdots\!16}{26\!\cdots\!37}a^{5}+\frac{13\!\cdots\!20}{26\!\cdots\!37}a^{4}-\frac{36\!\cdots\!24}{26\!\cdots\!37}a^{3}-\frac{17\!\cdots\!98}{26\!\cdots\!37}a^{2}+\frac{16\!\cdots\!82}{26\!\cdots\!37}a+\frac{72\!\cdots\!91}{26\!\cdots\!37}$, $\frac{174241811886310}{26\!\cdots\!37}a^{9}-\frac{314766711146625}{26\!\cdots\!37}a^{8}-\frac{21\!\cdots\!50}{26\!\cdots\!37}a^{7}+\frac{31\!\cdots\!85}{26\!\cdots\!37}a^{6}+\frac{10\!\cdots\!70}{26\!\cdots\!37}a^{5}-\frac{10\!\cdots\!10}{26\!\cdots\!37}a^{4}-\frac{24\!\cdots\!80}{26\!\cdots\!37}a^{3}+\frac{14\!\cdots\!35}{26\!\cdots\!37}a^{2}+\frac{36\!\cdots\!15}{26\!\cdots\!37}a+\frac{56\!\cdots\!58}{26\!\cdots\!37}$
| 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: | \( 601601.137232 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 601601.137232 \cdot 1}{2\cdot\sqrt{1412799778009275392}}\cr\approx \mathstrut & 0.259142294945 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279)
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^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279, 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^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279);
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^10 - 2*x^9 - 122*x^8 + 198*x^7 + 5582*x^6 - 6794*x^5 - 118785*x^4 + 94556*x^3 + 1168426*x^2 - 446344*x - 4240279);
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 cyclic group of order 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{23}) \), \(\Q(\zeta_{11})^+\) |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(23\)
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.92.2t1.a.a | $1$ | $ 2^{2} \cdot 23 $ | \(\Q(\sqrt{23}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.1012.10t1.a.a | $1$ | $ 2^{2} \cdot 11 \cdot 23 $ | 10.10.1412799778009275392.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.1012.10t1.a.b | $1$ | $ 2^{2} \cdot 11 \cdot 23 $ | 10.10.1412799778009275392.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.1012.10t1.a.c | $1$ | $ 2^{2} \cdot 11 \cdot 23 $ | 10.10.1412799778009275392.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.1012.10t1.a.d | $1$ | $ 2^{2} \cdot 11 \cdot 23 $ | 10.10.1412799778009275392.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.