Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41)
gp: K = bnfinit(y^12 - y^11 - 16*y^10 + 15*y^9 + 145*y^8 - 8*y^7 - 392*y^6 + 88*y^5 + 415*y^4 - 255*y^3 - 64*y^2 + 89*y - 41, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41)
\( x^{12} - x^{11} - 16 x^{10} + 15 x^{9} + 145 x^{8} - 8 x^{7} - 392 x^{6} + 88 x^{5} + 415 x^{4} + \cdots - 41 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1476225000000000000000000\)
\(\medspace = 2^{18}\cdot 3^{10}\cdot 5^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(103.30\) | 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: | $2^{13/6}3^{7/8}5^{39/20}\approx 270.8346732868732$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{5337160}a^{11}+\frac{65341}{2668580}a^{10}+\frac{8452}{133429}a^{9}+\frac{59635}{1067432}a^{8}+\frac{28320}{133429}a^{7}-\frac{219409}{2668580}a^{6}+\frac{49603}{667145}a^{5}-\frac{169855}{533716}a^{4}-\frac{91825}{1067432}a^{3}-\frac{20355}{133429}a^{2}+\frac{547919}{1334290}a-\frac{2562313}{5337160}$
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
$C_{2}$, which has order $2$ (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: | $7$ | 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: |
$\frac{40197}{1067432}a^{11}+\frac{95741}{533716}a^{10}-\frac{504661}{533716}a^{9}-\frac{2588317}{1067432}a^{8}+\frac{5260545}{533716}a^{7}+\frac{6641399}{266858}a^{6}-\frac{12426779}{533716}a^{5}-\frac{12324087}{266858}a^{4}+\frac{37011201}{1067432}a^{3}+\frac{2602727}{533716}a^{2}-\frac{1346640}{133429}a+\frac{5889469}{1067432}$, $\frac{2469919}{2668580}a^{11}-\frac{14146317}{2668580}a^{10}-\frac{157901}{133429}a^{9}+\frac{15417137}{266858}a^{8}-\frac{17845079}{533716}a^{7}-\frac{831744267}{2668580}a^{6}+\frac{1139219921}{2668580}a^{5}+\frac{150448883}{533716}a^{4}-\frac{257634533}{266858}a^{3}+\frac{221273377}{266858}a^{2}-\frac{928368891}{2668580}a+\frac{200171393}{2668580}$, $\frac{10973341}{5337160}a^{11}-\frac{19691659}{2668580}a^{10}-\frac{4309551}{266858}a^{9}+\frac{92464927}{1067432}a^{8}+\frac{16551379}{266858}a^{7}-\frac{693355799}{2668580}a^{6}-\frac{14602289}{1334290}a^{5}+\frac{159117979}{533716}a^{4}-\frac{154293913}{1067432}a^{3}-\frac{15761575}{266858}a^{2}+\frac{85878469}{1334290}a-\frac{140152193}{5337160}$, $\frac{7184123}{5337160}a^{11}-\frac{5062711}{1334290}a^{10}-\frac{7850627}{533716}a^{9}+\frac{50036315}{1067432}a^{8}+\frac{14900202}{133429}a^{7}-\frac{572830797}{2668580}a^{6}-\frac{101520331}{667145}a^{5}+\frac{212381681}{533716}a^{4}-\frac{153221795}{1067432}a^{3}-\frac{51811181}{533716}a^{2}+\frac{229912629}{2668580}a-\frac{169628129}{5337160}$, $\frac{470529}{1334290}a^{11}-\frac{416727}{1334290}a^{10}-\frac{3180091}{533716}a^{9}+\frac{1212505}{266858}a^{8}+\frac{29922905}{533716}a^{7}+\frac{14235571}{2668580}a^{6}-\frac{471685473}{2668580}a^{5}-\frac{23619285}{533716}a^{4}+\frac{96441587}{533716}a^{3}+\frac{1988925}{533716}a^{2}-\frac{31628701}{1334290}a+\frac{75096661}{2668580}$, $\frac{16233643}{5337160}a^{11}+\frac{9951703}{2668580}a^{10}-\frac{6028946}{133429}a^{9}-\frac{63787683}{1067432}a^{8}+\frac{50781543}{133429}a^{7}+\frac{2406779773}{2668580}a^{6}+\frac{259619763}{1334290}a^{5}-\frac{318535611}{533716}a^{4}-\frac{85819083}{1067432}a^{3}+\frac{8987546}{133429}a^{2}-\frac{57681994}{667145}a-\frac{96400619}{5337160}$, $\frac{189291}{5337160}a^{11}-\frac{405069}{2668580}a^{10}-\frac{104999}{533716}a^{9}+\frac{1876533}{1067432}a^{8}+\frac{177035}{533716}a^{7}-\frac{12194277}{1334290}a^{6}+\frac{30032497}{2668580}a^{5}+\frac{2058909}{133429}a^{4}-\frac{40412661}{1067432}a^{3}+\frac{3913153}{533716}a^{2}+\frac{31128109}{1334290}a-\frac{63080973}{5337160}$
| 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: | \( 279154681.275 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 279154681.275 \cdot 2}{2\cdot\sqrt{1476225000000000000000000}}\cr\approx \mathstrut & 5.72938614221 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41)
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 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41, 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 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41);
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 - x^11 - 16*x^10 + 15*x^9 + 145*x^8 - 8*x^7 - 392*x^6 + 88*x^5 + 415*x^4 - 255*x^3 - 64*x^2 + 89*x - 41);
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 non-solvable group of order 7920 |
| The 10 conjugacy class representatives for $M_{11}$ |
| Character table for $M_{11}$ |
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 11 sibling: | data not computed |
| Degree 22 sibling: | data not computed |
| Minimal sibling: | 11.3.6561000000000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.4.8.7 | $x^{4} + 4 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.6.10.1 | $x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.10.19.3 | $x^{10} + 50 x^{2} + 105$ | $10$ | $1$ | $19$ | $F_5$ | $[9/4]_{4}$ |