Normalized defining polynomial
\( x^{11} - 4312x^{4} + 614656 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-25363081849926339178101342208\) \(\medspace = -\,2^{17}\cdot 7^{14}\cdot 11^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(382.12\) | 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: | $2^{87/28}7^{25/14}11^{119/110}\approx 3724.2976664574426$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-22}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{14}a^{4}$, $\frac{1}{28}a^{5}$, $\frac{1}{28}a^{6}$, $\frac{1}{392}a^{7}$, $\frac{1}{784}a^{8}-\frac{1}{2}a$, $\frac{1}{10976}a^{9}-\frac{11}{28}a^{2}$, $\frac{1}{3973312}a^{10}+\frac{1}{124166}a^{9}-\frac{41}{141904}a^{8}+\frac{17}{17738}a^{7}+\frac{13}{1267}a^{6}+\frac{5}{724}a^{5}+\frac{17}{2534}a^{4}+\frac{717}{10136}a^{3}+\frac{334}{1267}a^{2}-\frac{75}{362}a+\frac{67}{181}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
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: | $\frac{383083185362519}{3973312}a^{10}-\frac{558667788865591}{1986656}a^{9}-\frac{67203993910399}{141904}a^{8}+\frac{78651583360715}{17738}a^{7}-\frac{3530855615803}{724}a^{6}-\frac{206716937105923}{5068}a^{5}+\frac{389630733082567}{2534}a^{4}-\frac{30\!\cdots\!65}{10136}a^{3}-\frac{44\!\cdots\!43}{5068}a^{2}+\frac{20\!\cdots\!85}{362}a-\frac{506702438214663}{181}$, $\frac{26\!\cdots\!23}{3973312}a^{10}+\frac{835274985232209}{1986656}a^{9}-\frac{233881206221529}{35476}a^{8}-\frac{636912538073677}{17738}a^{7}-\frac{101881934210471}{1267}a^{6}+\frac{238218874344791}{2534}a^{5}+\frac{18\!\cdots\!17}{1267}a^{4}+\frac{28\!\cdots\!35}{10136}a^{3}+\frac{24\!\cdots\!33}{5068}a^{2}-\frac{23\!\cdots\!46}{181}a-\frac{22\!\cdots\!49}{181}$, $\frac{46\!\cdots\!43}{81088}a^{10}-\frac{26\!\cdots\!15}{993328}a^{9}-\frac{63\!\cdots\!91}{141904}a^{8}+\frac{11\!\cdots\!59}{35476}a^{7}-\frac{91\!\cdots\!48}{1267}a^{6}-\frac{10\!\cdots\!19}{5068}a^{5}+\frac{34\!\cdots\!91}{2534}a^{4}-\frac{45\!\cdots\!35}{1448}a^{3}-\frac{73\!\cdots\!15}{2534}a^{2}+\frac{15\!\cdots\!31}{362}a-\frac{12\!\cdots\!25}{181}$, $\frac{1589477976575}{567616}a^{10}-\frac{4739203051645}{283808}a^{9}+\frac{904558190311}{141904}a^{8}+\frac{3226613196339}{17738}a^{7}-\frac{2556264368751}{5068}a^{6}-\frac{4841234587681}{5068}a^{5}+\frac{10392982953279}{1267}a^{4}-\frac{29196559285045}{1448}a^{3}-\frac{4094218764861}{724}a^{2}+\frac{90917275454351}{362}a-\frac{95189519971733}{181}$, $\frac{166982797375843}{993328}a^{10}-\frac{576216892695393}{1986656}a^{9}-\frac{183371169289853}{141904}a^{8}+\frac{116037627310791}{17738}a^{7}+\frac{297192367825}{5068}a^{6}-\frac{195252128693221}{2534}a^{5}+\frac{226711136583740}{1267}a^{4}-\frac{637295527432547}{2534}a^{3}-\frac{10\!\cdots\!61}{5068}a^{2}+\frac{27\!\cdots\!27}{362}a+\frac{10\!\cdots\!19}{181}$ (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: | \( 13573685854.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 13573685854.2 \cdot 2}{2\cdot\sqrt{25363081849926339178101342208}}\cr\approx \mathstrut & 1.66926819838 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ |
Character table for $S_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 22 sibling: | data not computed |
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 | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | R | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.11.12 | $x^{4} + 26$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.7.12.4 | $x^{7} + 42 x^{6} + 154$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ | |
\(11\) | 11.11.11.2 | $x^{11} + 88 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |