Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761)
gp: K = bnfinit(y^12 - 4*y^11 + 343*y^10 - 1126*y^9 + 47714*y^8 - 124252*y^7 + 3418417*y^6 - 6656386*y^5 + 132905105*y^4 - 173414484*y^3 + 2704663170*y^2 - 1800348698*y + 22837898761, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761)
\( x^{12} - 4 x^{11} + 343 x^{10} - 1126 x^{9} + 47714 x^{8} - 124252 x^{7} + 3418417 x^{6} + \cdots + 22837898761 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6827179527544712000000000\)
\(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(117.36\) | 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: | $2\cdot 5^{3/4}7^{2/3}23^{1/2}\approx 117.36000254893146$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3220=2^{2}\cdot 5\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3220}(1,·)$, $\chi_{3220}(1563,·)$, $\chi_{3220}(1381,·)$, $\chi_{3220}(2209,·)$, $\chi_{3220}(1289,·)$, $\chi_{3220}(2669,·)$, $\chi_{3220}(1103,·)$, $\chi_{3220}(1747,·)$, $\chi_{3220}(183,·)$, $\chi_{3220}(921,·)$, $\chi_{3220}(827,·)$, $\chi_{3220}(2207,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.1058000.1$^{2}$, 12.0.6827179527544712000000000.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23}a^{6}-\frac{2}{23}a^{5}-\frac{3}{23}a^{4}+\frac{6}{23}a^{3}+\frac{2}{23}a^{2}-\frac{4}{23}a+\frac{1}{23}$, $\frac{1}{23}a^{7}-\frac{7}{23}a^{5}-\frac{9}{23}a^{3}-\frac{7}{23}a+\frac{2}{23}$, $\frac{1}{23}a^{8}+\frac{9}{23}a^{5}-\frac{7}{23}a^{4}-\frac{4}{23}a^{3}+\frac{7}{23}a^{2}-\frac{3}{23}a+\frac{7}{23}$, $\frac{1}{23}a^{9}+\frac{11}{23}a^{5}-\frac{1}{23}a^{3}+\frac{2}{23}a^{2}-\frac{3}{23}a-\frac{9}{23}$, $\frac{1}{1139352172103}a^{10}-\frac{20806941616}{1139352172103}a^{9}+\frac{6087284113}{1139352172103}a^{8}+\frac{16492897723}{1139352172103}a^{7}-\frac{16686069814}{1139352172103}a^{6}-\frac{372442820778}{1139352172103}a^{5}+\frac{422114670793}{1139352172103}a^{4}+\frac{356886078144}{1139352172103}a^{3}-\frac{20470744049}{49537050961}a^{2}+\frac{276992754248}{1139352172103}a-\frac{400022280621}{1139352172103}$, $\frac{1}{10\!\cdots\!23}a^{11}-\frac{5248711016065}{10\!\cdots\!23}a^{10}+\frac{13\!\cdots\!24}{10\!\cdots\!23}a^{9}+\frac{13\!\cdots\!26}{10\!\cdots\!23}a^{8}-\frac{95\!\cdots\!20}{10\!\cdots\!23}a^{7}+\frac{48\!\cdots\!74}{43\!\cdots\!01}a^{6}-\frac{41\!\cdots\!43}{10\!\cdots\!23}a^{5}+\frac{13\!\cdots\!12}{10\!\cdots\!23}a^{4}+\frac{43\!\cdots\!42}{10\!\cdots\!23}a^{3}+\frac{85\!\cdots\!78}{10\!\cdots\!23}a^{2}+\frac{12\!\cdots\!96}{10\!\cdots\!23}a-\frac{53\!\cdots\!52}{10\!\cdots\!23}$
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
$C_{2}\times C_{62770}$, which has order $125540$ (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: | $5$ | 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: |
$\frac{28}{1139352172103}a^{11}+\frac{19020}{1139352172103}a^{10}-\frac{54720}{1139352172103}a^{9}+\frac{5343585}{1139352172103}a^{8}-\frac{13021300}{1139352172103}a^{7}+\frac{570769682}{1139352172103}a^{6}-\frac{1062410566}{1139352172103}a^{5}+\frac{28408642905}{1139352172103}a^{4}-\frac{35674206770}{1139352172103}a^{3}+\frac{687693433721}{1139352172103}a^{2}-\frac{443943927008}{1139352172103}a+\frac{8492669250761}{1139352172103}$, $\frac{89\!\cdots\!60}{43\!\cdots\!01}a^{11}-\frac{15\!\cdots\!60}{43\!\cdots\!01}a^{10}+\frac{28\!\cdots\!80}{43\!\cdots\!01}a^{9}-\frac{50\!\cdots\!55}{43\!\cdots\!01}a^{8}+\frac{35\!\cdots\!00}{43\!\cdots\!01}a^{7}-\frac{61\!\cdots\!50}{43\!\cdots\!01}a^{6}+\frac{21\!\cdots\!24}{43\!\cdots\!01}a^{5}-\frac{36\!\cdots\!75}{43\!\cdots\!01}a^{4}+\frac{61\!\cdots\!00}{43\!\cdots\!01}a^{3}-\frac{10\!\cdots\!00}{43\!\cdots\!01}a^{2}+\frac{70\!\cdots\!40}{43\!\cdots\!01}a-\frac{12\!\cdots\!90}{43\!\cdots\!01}$, $\frac{33\!\cdots\!00}{43\!\cdots\!01}a^{11}+\frac{11\!\cdots\!26}{43\!\cdots\!01}a^{10}+\frac{47\!\cdots\!60}{43\!\cdots\!01}a^{9}+\frac{40\!\cdots\!45}{43\!\cdots\!01}a^{8}-\frac{25\!\cdots\!80}{43\!\cdots\!01}a^{7}+\frac{54\!\cdots\!80}{43\!\cdots\!01}a^{6}-\frac{74\!\cdots\!10}{43\!\cdots\!01}a^{5}+\frac{34\!\cdots\!80}{43\!\cdots\!01}a^{4}-\frac{41\!\cdots\!90}{43\!\cdots\!01}a^{3}+\frac{10\!\cdots\!45}{43\!\cdots\!01}a^{2}-\frac{71\!\cdots\!20}{43\!\cdots\!01}a+\frac{12\!\cdots\!94}{43\!\cdots\!01}$, $\frac{28\!\cdots\!32}{10\!\cdots\!23}a^{11}-\frac{27\!\cdots\!02}{10\!\cdots\!23}a^{10}+\frac{80\!\cdots\!40}{10\!\cdots\!23}a^{9}-\frac{70\!\cdots\!15}{10\!\cdots\!23}a^{8}+\frac{86\!\cdots\!60}{10\!\cdots\!23}a^{7}-\frac{68\!\cdots\!72}{10\!\cdots\!23}a^{6}+\frac{41\!\cdots\!28}{10\!\cdots\!23}a^{5}-\frac{30\!\cdots\!90}{10\!\cdots\!23}a^{4}+\frac{77\!\cdots\!00}{10\!\cdots\!23}a^{3}-\frac{66\!\cdots\!26}{10\!\cdots\!23}a^{2}+\frac{37\!\cdots\!88}{10\!\cdots\!23}a-\frac{72\!\cdots\!09}{10\!\cdots\!23}$, $\frac{74\!\cdots\!52}{10\!\cdots\!23}a^{11}+\frac{93\!\cdots\!78}{10\!\cdots\!23}a^{10}+\frac{15\!\cdots\!00}{10\!\cdots\!23}a^{9}+\frac{45\!\cdots\!50}{10\!\cdots\!23}a^{8}+\frac{55\!\cdots\!60}{10\!\cdots\!23}a^{7}+\frac{74\!\cdots\!78}{10\!\cdots\!23}a^{6}-\frac{78\!\cdots\!24}{10\!\cdots\!23}a^{5}+\frac{53\!\cdots\!35}{10\!\cdots\!23}a^{4}-\frac{64\!\cdots\!00}{10\!\cdots\!23}a^{3}+\frac{17\!\cdots\!74}{10\!\cdots\!23}a^{2}-\frac{12\!\cdots\!32}{10\!\cdots\!23}a+\frac{21\!\cdots\!61}{10\!\cdots\!23}$
| 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: | \( 104.882003477 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 125540}{2\cdot\sqrt{6827179527544712000000000}}\cr\approx \mathstrut & 0.155028538181 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761)
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^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761, 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^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761);
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^12 - 4*x^11 + 343*x^10 - 1126*x^9 + 47714*x^8 - 124252*x^7 + 3418417*x^6 - 6656386*x^5 + 132905105*x^4 - 173414484*x^3 + 2704663170*x^2 - 1800348698*x + 22837898761);
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 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1058000.1, 6.6.300125.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
|
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(23\)
| 23.12.6.2 | $x^{12} + 529 x^{8} - 109503 x^{6} + 2518569 x^{4} - 6436343 x^{2} + 740179445$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |