Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 1)
gp: K = bnfinit(y^23 + 23*y^21 + 230*y^19 + 1311*y^17 + 4692*y^15 + 10948*y^13 + 16744*y^11 + 16445*y^9 + 9867*y^7 + 3289*y^5 + 506*y^3 + 23*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 1)
\( x^{23} + 23 x^{21} + 230 x^{19} + 1311 x^{17} + 4692 x^{15} + 10948 x^{13} + 16744 x^{11} + 16445 x^{9} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1019554101555073829802491841336279296875\)
\(\medspace = -\,5^{11}\cdot 23^{23}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.66\) | 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: | $5^{1/2}23^{527/506}\approx 58.57701468324603$ | ||
| Ramified primes: |
\(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-115}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
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: | $11$ | 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: |
$a$, $2a^{21}+42a^{19}+378a^{17}+1904a^{15}+5880a^{13}+11466a^{11}+14014a^{9}+10296a^{7}+4158a^{5}+769a^{3}-a^{2}+37a-2$, $a^{20}+a^{19}+22a^{18}+20a^{17}+207a^{16}+168a^{15}+1087a^{14}+767a^{13}+3486a^{12}+2054a^{11}+7019a^{10}+3236a^{9}+8776a^{8}+2824a^{7}+6472a^{6}+1150a^{5}+2489a^{4}+119a^{3}+356a^{2}-4a-1$, $a^{17}-a^{16}+16a^{15}-16a^{14}+104a^{13}-104a^{12}+351a^{11}-351a^{10}+651a^{9}-651a^{8}+645a^{7}-645a^{6}+307a^{5}-307a^{4}+58a^{3}-58a^{2}+2a-3$, $a^{20}-a^{19}+19a^{18}-19a^{17}+153a^{16}-153a^{15}+680a^{14}-680a^{13}+1819a^{12}-1819a^{11}+2992a^{10}-2992a^{9}+2957a^{8}-2957a^{7}+1625a^{6}-1625a^{5}+410a^{4}-410a^{3}+24a^{2}-24a-1$, $a^{22}+24a^{20}+249a^{18}+1462a^{16}+5340a^{14}-a^{13}+12557a^{12}-14a^{11}+19007a^{10}-76a^{9}+17964a^{8}-201a^{7}+9886a^{6}-266a^{5}+2749a^{4}-159a^{3}+276a^{2}-28a+1$, $a^{19}-a^{18}+18a^{17}-18a^{16}+134a^{15}-134a^{14}+533a^{13}-533a^{12}+1223a^{11}-1223a^{10}+1635a^{9}-1635a^{8}+1231a^{7}-1231a^{6}+480a^{5}-480a^{4}+81a^{3}-81a^{2}+3a-4$, $a^{21}+2a^{20}+20a^{19}+39a^{18}+171a^{17}+324a^{16}+817a^{15}+1498a^{14}+2393a^{13}+4226a^{12}+4435a^{11}+7504a^{10}+5180a^{9}+8347a^{8}+3678a^{7}+5595a^{6}+1471a^{5}+2065a^{4}+290a^{3}+343a^{2}+24a+16$, $a^{20}-a^{19}+18a^{18}-18a^{17}+136a^{16}-136a^{15}+560a^{14}-561a^{13}+1365a^{12}-1378a^{11}+2003a^{10}-2068a^{9}+1725a^{8}-1879a^{7}+821a^{6}-989a^{5}+204a^{4}-267a^{3}+29a^{2}-27a+3$, $2a^{22}+4a^{21}+43a^{20}+87a^{19}+396a^{18}+808a^{17}+2039a^{16}+4177a^{15}+6429a^{14}+13131a^{13}+12794a^{12}+25747a^{11}+16013a^{10}+31074a^{9}+12237a^{8}+21977a^{7}+5393a^{6}+8283a^{5}+1233a^{4}+1389a^{3}+117a^{2}+68a-2$, $a^{22}+a^{21}+22a^{20}+20a^{19}+210a^{18}+171a^{17}+1140a^{16}+817a^{15}+3876a^{14}+2394a^{13}+8567a^{12}+4447a^{11}+12363a^{10}+5236a^{9}+11376a^{8}+3807a^{7}+6287a^{6}+1623a^{5}+1839a^{4}+373a^{3}+207a^{2}+39a-1$
| 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: | \( 53144954646.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 53144954646.2 \cdot 1}{2\cdot\sqrt{1019554101555073829802491841336279296875}}\cr\approx \mathstrut & 1.00284878958 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 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^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 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^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 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^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x - 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
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 506 |
| The 23 conjugacy class representatives for $F_{23}$ |
| Character table for $F_{23}$ 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]
Sibling fields
| Degree 46 sibling: | 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 | $22{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.11.0.1}{11} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.11.0.1}{11} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{11}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.22.11.2 | $x^{22} + 29296875 x^{2} - 146484375$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ | |
|
\(23\)
| 23.23.23.11 | $x^{23} + 506 x + 23$ | $23$ | $1$ | $23$ | $F_{23}$ | $[23/22]_{22}$ |