Normalized defining polynomial
\( x^{12} - 4 x^{11} - 286 x^{9} + 1287 x^{8} - 2288 x^{6} - 1716 x^{5} - 15873 x^{4} + 2860 x^{3} + \cdots - 16159 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-40276619896167543142132736\) \(\medspace = -\,2^{10}\cdot 11^{11}\cdot 13^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(136.07\) | 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: | $2^{10/11}11^{119/110}13^{10/11}\approx 258.7837181718594$ | ||
Ramified primes: | \(2\), \(11\), \(13\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{2}$, $\frac{1}{16\!\cdots\!68}a^{11}-\frac{13\!\cdots\!15}{16\!\cdots\!68}a^{10}-\frac{15\!\cdots\!19}{16\!\cdots\!68}a^{9}-\frac{85\!\cdots\!49}{16\!\cdots\!68}a^{8}+\frac{32\!\cdots\!81}{83\!\cdots\!84}a^{7}+\frac{19\!\cdots\!95}{83\!\cdots\!84}a^{6}+\frac{17\!\cdots\!49}{83\!\cdots\!84}a^{5}-\frac{11\!\cdots\!11}{83\!\cdots\!84}a^{4}-\frac{17\!\cdots\!63}{16\!\cdots\!68}a^{3}+\frac{79\!\cdots\!81}{16\!\cdots\!68}a^{2}+\frac{64\!\cdots\!57}{16\!\cdots\!68}a+\frac{81\!\cdots\!95}{16\!\cdots\!68}$
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)
| |
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: | $\frac{45\!\cdots\!59}{41\!\cdots\!42}a^{11}-\frac{24\!\cdots\!15}{41\!\cdots\!42}a^{10}+\frac{82\!\cdots\!91}{41\!\cdots\!42}a^{9}-\frac{25\!\cdots\!89}{83\!\cdots\!84}a^{8}+\frac{37\!\cdots\!02}{20\!\cdots\!71}a^{7}-\frac{16\!\cdots\!84}{20\!\cdots\!71}a^{6}-\frac{14\!\cdots\!78}{20\!\cdots\!71}a^{5}-\frac{38\!\cdots\!59}{20\!\cdots\!71}a^{4}-\frac{76\!\cdots\!41}{41\!\cdots\!42}a^{3}+\frac{18\!\cdots\!47}{41\!\cdots\!42}a^{2}+\frac{17\!\cdots\!99}{41\!\cdots\!42}a+\frac{18\!\cdots\!73}{83\!\cdots\!84}$, $\frac{29\!\cdots\!91}{41\!\cdots\!42}a^{11}-\frac{10\!\cdots\!97}{83\!\cdots\!84}a^{10}-\frac{98\!\cdots\!89}{83\!\cdots\!84}a^{9}-\frac{15\!\cdots\!21}{83\!\cdots\!84}a^{8}+\frac{91\!\cdots\!33}{20\!\cdots\!71}a^{7}+\frac{72\!\cdots\!33}{20\!\cdots\!71}a^{6}-\frac{15\!\cdots\!79}{20\!\cdots\!71}a^{5}-\frac{25\!\cdots\!49}{41\!\cdots\!42}a^{4}+\frac{45\!\cdots\!65}{41\!\cdots\!42}a^{3}-\frac{35\!\cdots\!99}{83\!\cdots\!84}a^{2}+\frac{60\!\cdots\!09}{83\!\cdots\!84}a-\frac{15\!\cdots\!69}{83\!\cdots\!84}$, $\frac{15\!\cdots\!17}{83\!\cdots\!84}a^{11}-\frac{68\!\cdots\!47}{83\!\cdots\!84}a^{10}-\frac{99\!\cdots\!93}{83\!\cdots\!84}a^{9}-\frac{10\!\cdots\!26}{20\!\cdots\!71}a^{8}+\frac{10\!\cdots\!49}{41\!\cdots\!42}a^{7}+\frac{91\!\cdots\!75}{41\!\cdots\!42}a^{6}-\frac{43\!\cdots\!65}{41\!\cdots\!42}a^{5}+\frac{17\!\cdots\!73}{41\!\cdots\!42}a^{4}-\frac{11\!\cdots\!19}{83\!\cdots\!84}a^{3}-\frac{15\!\cdots\!55}{83\!\cdots\!84}a^{2}-\frac{30\!\cdots\!01}{83\!\cdots\!84}a-\frac{63\!\cdots\!29}{41\!\cdots\!42}$, $\frac{41\!\cdots\!93}{83\!\cdots\!84}a^{11}-\frac{44\!\cdots\!59}{41\!\cdots\!42}a^{10}+\frac{20\!\cdots\!89}{83\!\cdots\!84}a^{9}-\frac{19\!\cdots\!19}{20\!\cdots\!71}a^{8}+\frac{13\!\cdots\!75}{41\!\cdots\!42}a^{7}-\frac{34\!\cdots\!33}{41\!\cdots\!42}a^{6}-\frac{33\!\cdots\!75}{20\!\cdots\!71}a^{5}+\frac{39\!\cdots\!37}{20\!\cdots\!71}a^{4}+\frac{49\!\cdots\!45}{83\!\cdots\!84}a^{3}+\frac{36\!\cdots\!85}{20\!\cdots\!71}a^{2}+\frac{25\!\cdots\!15}{83\!\cdots\!84}a+\frac{16\!\cdots\!78}{20\!\cdots\!71}$, $\frac{11\!\cdots\!51}{83\!\cdots\!84}a^{11}-\frac{10\!\cdots\!57}{83\!\cdots\!84}a^{10}-\frac{12\!\cdots\!07}{83\!\cdots\!84}a^{9}-\frac{32\!\cdots\!49}{83\!\cdots\!84}a^{8}+\frac{23\!\cdots\!39}{41\!\cdots\!42}a^{7}+\frac{20\!\cdots\!63}{41\!\cdots\!42}a^{6}-\frac{22\!\cdots\!64}{20\!\cdots\!71}a^{5}-\frac{62\!\cdots\!97}{41\!\cdots\!42}a^{4}-\frac{37\!\cdots\!97}{83\!\cdots\!84}a^{3}-\frac{40\!\cdots\!85}{83\!\cdots\!84}a^{2}-\frac{17\!\cdots\!69}{83\!\cdots\!84}a-\frac{15\!\cdots\!65}{83\!\cdots\!84}$, $\frac{29\!\cdots\!69}{83\!\cdots\!84}a^{11}-\frac{17\!\cdots\!13}{83\!\cdots\!84}a^{10}+\frac{36\!\cdots\!47}{83\!\cdots\!84}a^{9}-\frac{91\!\cdots\!73}{83\!\cdots\!84}a^{8}+\frac{28\!\cdots\!25}{41\!\cdots\!42}a^{7}-\frac{29\!\cdots\!37}{20\!\cdots\!71}a^{6}+\frac{86\!\cdots\!89}{41\!\cdots\!42}a^{5}-\frac{10\!\cdots\!31}{20\!\cdots\!71}a^{4}+\frac{36\!\cdots\!37}{83\!\cdots\!84}a^{3}-\frac{66\!\cdots\!43}{83\!\cdots\!84}a^{2}+\frac{14\!\cdots\!15}{83\!\cdots\!84}a-\frac{23\!\cdots\!87}{83\!\cdots\!84}$ (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)]
| |
Regulator: | \( 580319347.562 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 580319347.562 \cdot 1}{2\cdot\sqrt{40276619896167543142132736}}\cr\approx \mathstrut & 1.79089399072 \end{aligned}\] (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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | R | R | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.11.10.1 | $x^{11} + 2$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.11.11.10 | $x^{11} + 33 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.11.10.1 | $x^{11} + 13$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ |
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) | $1$ | $-1$ | |
10.402...736.55.a.a | $10$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.402...736.55.b.a | $10$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.402...736.110.a.a | $10$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.402...736.22t14.a.a | $10$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
10.402...736.22t14.a.b | $10$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
* | 11.402...736.12t218.a.a | $11$ | $ 2^{10} \cdot 11^{11} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $1$ |
11.443...096.24t2949.a.a | $11$ | $ 2^{10} \cdot 11^{12} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-1$ | |
12.487...056.55.a.a | $12$ | $ 2^{10} \cdot 11^{13} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ | |
12.487...056.110.a.a | $12$ | $ 2^{10} \cdot 11^{13} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
12.487...056.110.a.b | $12$ | $ 2^{10} \cdot 11^{13} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
12.487...056.55.a.b | $12$ | $ 2^{10} \cdot 11^{13} \cdot 13^{10}$ | 12.2.40276619896167543142132736.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ |