Normalized defining polynomial
\( x^{11} - 33 x^{9} - 176 x^{8} - 1881 x^{7} - 9768 x^{6} - 96962 x^{5} - 970164 x^{4} - 4535019 x^{3} + \cdots - 3808296 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4325187603056652501797342844511965821771743681\) \(\medspace = 19^{4}\cdot 3659^{4}\cdot 3688801^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14\,084.06\) | 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: | $19^{1/2}3659^{1/2}3688801^{1/2}\approx 506408.0709477289$ | ||
Ramified primes: | \(19\), \(3659\), \(3688801\) | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{36}a^{5}+\frac{1}{36}a^{4}+\frac{1}{36}a^{3}-\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{108}a^{6}-\frac{13}{108}a^{3}-\frac{5}{36}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{108}a^{7}-\frac{1}{108}a^{4}-\frac{1}{36}a^{3}-\frac{5}{36}a^{2}+\frac{1}{6}a$, $\frac{1}{324}a^{8}-\frac{1}{324}a^{7}-\frac{1}{81}a^{5}+\frac{7}{324}a^{4}-\frac{1}{108}a^{3}-\frac{1}{12}a^{2}+\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{648}a^{9}-\frac{1}{648}a^{8}+\frac{1}{324}a^{6}+\frac{7}{648}a^{5}+\frac{11}{216}a^{4}+\frac{13}{216}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{3888}a^{10}-\frac{1}{1944}a^{9}-\frac{5}{3888}a^{8}+\frac{1}{1944}a^{7}+\frac{11}{3888}a^{6}+\frac{7}{1944}a^{5}-\frac{11}{648}a^{4}-\frac{7}{108}a^{3}-\frac{1}{144}a^{2}-\frac{5}{12}a-\frac{1}{4}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{11\!\cdots\!41}{3888}a^{10}-\frac{15\!\cdots\!87}{1944}a^{9}-\frac{28\!\cdots\!89}{3888}a^{8}-\frac{54\!\cdots\!05}{1944}a^{7}-\frac{16\!\cdots\!57}{3888}a^{6}-\frac{29\!\cdots\!51}{1944}a^{5}-\frac{15\!\cdots\!11}{648}a^{4}-\frac{22\!\cdots\!43}{108}a^{3}-\frac{96\!\cdots\!73}{144}a^{2}-\frac{10\!\cdots\!23}{12}a-\frac{14\!\cdots\!17}{4}$, $\frac{38\!\cdots\!97}{648}a^{10}+\frac{48\!\cdots\!66}{81}a^{9}+\frac{26\!\cdots\!59}{648}a^{8}+\frac{10\!\cdots\!17}{324}a^{7}+\frac{13\!\cdots\!55}{648}a^{6}+\frac{23\!\cdots\!97}{162}a^{5}+\frac{48\!\cdots\!81}{54}a^{4}+\frac{35\!\cdots\!55}{108}a^{3}+\frac{52\!\cdots\!39}{8}a^{2}+\frac{25\!\cdots\!39}{4}a+22\!\cdots\!04$, $\frac{89\!\cdots\!61}{1296}a^{10}-\frac{43\!\cdots\!51}{216}a^{9}-\frac{21\!\cdots\!89}{1296}a^{8}-\frac{46\!\cdots\!13}{648}a^{7}-\frac{47\!\cdots\!79}{432}a^{6}-\frac{23\!\cdots\!79}{648}a^{5}-\frac{40\!\cdots\!09}{72}a^{4}-\frac{91\!\cdots\!29}{18}a^{3}-\frac{26\!\cdots\!19}{16}a^{2}-\frac{87\!\cdots\!51}{4}a-\frac{36\!\cdots\!87}{4}$, $\frac{91\!\cdots\!25}{648}a^{10}-\frac{35\!\cdots\!02}{81}a^{9}-\frac{20\!\cdots\!33}{648}a^{8}-\frac{46\!\cdots\!03}{324}a^{7}-\frac{14\!\cdots\!23}{648}a^{6}-\frac{22\!\cdots\!85}{324}a^{5}-\frac{37\!\cdots\!21}{324}a^{4}-\frac{10\!\cdots\!15}{108}a^{3}-\frac{23\!\cdots\!71}{72}a^{2}-\frac{49\!\cdots\!01}{12}a-16\!\cdots\!90$, $\frac{64\!\cdots\!17}{3888}a^{10}-\frac{16\!\cdots\!27}{1944}a^{9}-\frac{56\!\cdots\!97}{3888}a^{8}-\frac{43\!\cdots\!41}{1944}a^{7}-\frac{79\!\cdots\!77}{3888}a^{6}-\frac{12\!\cdots\!03}{1944}a^{5}-\frac{31\!\cdots\!49}{24}a^{4}-\frac{10\!\cdots\!69}{108}a^{3}-\frac{40\!\cdots\!61}{144}a^{2}-\frac{99\!\cdots\!62}{3}a-\frac{51\!\cdots\!07}{4}$, $\frac{55\!\cdots\!11}{972}a^{10}+\frac{55\!\cdots\!23}{972}a^{9}+\frac{19\!\cdots\!51}{486}a^{8}+\frac{28\!\cdots\!85}{972}a^{7}+\frac{18\!\cdots\!23}{972}a^{6}+\frac{13\!\cdots\!15}{972}a^{5}+\frac{13\!\cdots\!83}{162}a^{4}+\frac{95\!\cdots\!69}{3}a^{3}+\frac{22\!\cdots\!53}{36}a^{2}+\frac{12\!\cdots\!49}{2}a+21\!\cdots\!49$ (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: | \( 44689116249000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 44689116249000000000 \cdot 1}{2\cdot\sqrt{4325187603056652501797342844511965821771743681}}\cr\approx \mathstrut & 4.23622243912647 \end{aligned}\] (assuming GRH)
Galois group
$\PSL(2,11)$ (as 11T5):
A non-solvable group of order 660 |
The 8 conjugacy class representatives for $\PSL(2,11)$ |
Character table for $\PSL(2,11)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 12 sibling: | 12.0.284450973418865813541312928543110108576585288434610303751085363701921.1 |
Arithmetically equvalently sibling: | 11.3.4325187603056652501797342844511965821771743681.1 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(3659\) | $\Q_{3659}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
\(3688801\) | $\Q_{3688801}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
5.657...041.55.a.a | $5$ | $ 19^{2} \cdot 3659^{2} \cdot 3688801^{2}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $0$ | $1$ | |
5.657...041.55.a.b | $5$ | $ 19^{2} \cdot 3659^{2} \cdot 3688801^{2}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $0$ | $1$ | |
* | 10.432...681.11t5.a.a | $10$ | $ 19^{4} \cdot 3659^{4} \cdot 3688801^{4}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $1$ | $2$ |
10.284...921.55.a.a | $10$ | $ 19^{6} \cdot 3659^{6} \cdot 3688801^{6}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $1$ | $-2$ | |
11.284...921.12t179.a.a | $11$ | $ 19^{6} \cdot 3659^{6} \cdot 3688801^{6}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $1$ | $-1$ | |
12.284...921.55.a.a | $12$ | $ 19^{6} \cdot 3659^{6} \cdot 3688801^{6}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $1$ | $0$ | |
12.284...921.55.a.b | $12$ | $ 19^{6} \cdot 3659^{6} \cdot 3688801^{6}$ | 11.3.4325187603056652501797342844511965821771743681.2 | $\PSL(2,11)$ (as 11T5) | $1$ | $0$ |