Normalized defining polynomial
\( x^{12} - 2x^{11} - 99x^{8} + 132x^{7} - 132x^{6} + 3267x^{4} - 2178x^{3} + 4356x^{2} - 1656x - 35793 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-276027291040300056576\) \(\medspace = -\,2^{14}\cdot 3^{10}\cdot 11^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(50.51\) | 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))
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Galois root discriminant: | $2^{7/6}3^{10/11}11^{119/110}\approx 81.57288043386542$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{84}a^{7}-\frac{5}{84}a^{6}+\frac{3}{28}a^{5}-\frac{1}{28}a^{4}+\frac{1}{4}a^{3}+\frac{9}{28}a^{2}-\frac{5}{28}a-\frac{1}{28}$, $\frac{1}{168}a^{8}-\frac{2}{21}a^{6}+\frac{1}{28}a^{4}+\frac{2}{7}a^{3}+\frac{3}{14}a^{2}+\frac{2}{7}a-\frac{5}{56}$, $\frac{1}{168}a^{9}+\frac{1}{42}a^{6}-\frac{3}{28}a^{5}+\frac{3}{14}a^{3}+\frac{5}{14}a^{2}+\frac{27}{56}a-\frac{2}{7}$, $\frac{1}{168}a^{10}+\frac{1}{84}a^{6}-\frac{3}{14}a^{5}-\frac{3}{14}a^{4}-\frac{1}{7}a^{3}-\frac{9}{56}a^{2}+\frac{1}{14}a-\frac{3}{7}$, $\frac{1}{1306704}a^{11}+\frac{3239}{1306704}a^{10}-\frac{2701}{1306704}a^{9}-\frac{3691}{1306704}a^{8}-\frac{11}{217784}a^{7}-\frac{3047}{93336}a^{6}-\frac{48091}{217784}a^{5}-\frac{54035}{217784}a^{4}-\frac{134889}{435568}a^{3}+\frac{180965}{435568}a^{2}+\frac{172265}{435568}a-\frac{111197}{435568}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{161}{186672}a^{11}-\frac{20863}{1306704}a^{10}+\frac{36061}{1306704}a^{9}+\frac{110365}{1306704}a^{8}-\frac{184973}{653352}a^{7}+\frac{727369}{653352}a^{6}-\frac{149235}{217784}a^{5}-\frac{967955}{217784}a^{4}+\frac{3866773}{435568}a^{3}-\frac{10974141}{435568}a^{2}-\frac{3402989}{435568}a+\frac{37606747}{435568}$, $\frac{34}{11667}a^{11}-\frac{449}{326676}a^{10}-\frac{5335}{653352}a^{9}+\frac{152}{27223}a^{8}-\frac{66721}{326676}a^{7}-\frac{3364}{27223}a^{6}-\frac{4551}{27223}a^{5}-\frac{29926}{27223}a^{4}+\frac{346573}{108892}a^{3}+\frac{559885}{108892}a^{2}+\frac{261971}{31112}a+\frac{843431}{54446}$, $\frac{755}{435568}a^{11}+\frac{1681}{1306704}a^{10}+\frac{503}{186672}a^{9}+\frac{9013}{1306704}a^{8}-\frac{90301}{653352}a^{7}-\frac{187699}{653352}a^{6}-\frac{244121}{217784}a^{5}-\frac{414679}{217784}a^{4}+\frac{805167}{435568}a^{3}+\frac{441221}{62224}a^{2}+\frac{12270539}{435568}a+\frac{15048643}{435568}$, $\frac{1753}{108892}a^{11}-\frac{3727}{653352}a^{10}-\frac{23325}{217784}a^{9}+\frac{8751}{108892}a^{8}-\frac{26980}{27223}a^{7}-\frac{92728}{81669}a^{6}+\frac{390547}{108892}a^{5}-\frac{316679}{108892}a^{4}+\frac{2017963}{108892}a^{3}+\frac{10563013}{217784}a^{2}-\frac{8024837}{217784}a-\frac{3461201}{54446}$, $\frac{6935}{435568}a^{11}-\frac{78977}{1306704}a^{10}+\frac{32857}{1306704}a^{9}+\frac{10045}{62224}a^{8}-\frac{139193}{93336}a^{7}+\frac{2295121}{653352}a^{6}-\frac{422459}{217784}a^{5}-\frac{736077}{217784}a^{4}+\frac{15912767}{435568}a^{3}-\frac{3628305}{62224}a^{2}+\frac{7967823}{435568}a-\frac{20016949}{435568}$, $\frac{45649}{1306704}a^{11}-\frac{2669}{1306704}a^{10}-\frac{158517}{435568}a^{9}+\frac{7615}{62224}a^{8}-\frac{868621}{653352}a^{7}-\frac{2284213}{653352}a^{6}+\frac{3215643}{217784}a^{5}+\frac{1292351}{217784}a^{4}+\frac{10592711}{435568}a^{3}+\frac{60230321}{435568}a^{2}-\frac{90330045}{435568}a-\frac{206156051}{435568}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1255163.01641 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 1255163.01641 \cdot 1}{2\cdot\sqrt{276027291040300056576}}\cr\approx \mathstrut & 1.47963113059 \end{aligned}\] (assuming GRH)
Galois group
$\PGL(2,11)$ (as 12T218):
A non-solvable group of order 1320 |
The 13 conjugacy class representatives for $\PGL(2,11)$ |
Character table for $\PGL(2,11)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 22 sibling: | data not computed |
Degree 24 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.14.1 | $x^{12} + 2 x^{3} + 2$ | $12$ | $1$ | $14$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.11.10.1 | $x^{11} + 3$ | $11$ | $1$ | $10$ | $C_{11}:C_5$ | $[\ ]_{11}^{5}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.11.11.9 | $x^{11} + 99 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
10.690...144.55.a.a | $10$ | $ 2^{12} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.690...144.55.b.a | $10$ | $ 2^{12} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.690...144.110.a.a | $10$ | $ 2^{12} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.172...536.22t14.a.a | $10$ | $ 2^{10} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.172...536.22t14.a.b | $10$ | $ 2^{10} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
* | 11.276...576.12t218.a.a | $11$ | $ 2^{14} \cdot 3^{10} \cdot 11^{11}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $1$ |
11.303...336.24t2949.a.a | $11$ | $ 2^{14} \cdot 3^{10} \cdot 11^{12}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-1$ | |
12.333...696.55.a.a | $12$ | $ 2^{14} \cdot 3^{10} \cdot 11^{13}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ | |
12.333...696.110.a.a | $12$ | $ 2^{14} \cdot 3^{10} \cdot 11^{13}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
12.333...696.110.a.b | $12$ | $ 2^{14} \cdot 3^{10} \cdot 11^{13}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
12.333...696.55.a.b | $12$ | $ 2^{14} \cdot 3^{10} \cdot 11^{13}$ | 12.2.276027291040300056576.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ |