Normalized defining polynomial
\( x^{11} - 847 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2090387445666559894871442426539\) \(\medspace = -\,7^{10}\cdot 11^{21}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(570.67\) | 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: | $7^{10/11}11^{219/110}\approx 694.3670589522617$ | ||
Ramified primes: | \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11}a^{6}$, $\frac{1}{11}a^{7}$, $\frac{1}{11}a^{8}$, $\frac{1}{11}a^{9}$, $\frac{1}{11}a^{10}$
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: | $\frac{47\!\cdots\!50}{11}a^{10}-\frac{28\!\cdots\!57}{11}a^{9}-\frac{60\!\cdots\!44}{11}a^{8}+\frac{54\!\cdots\!04}{11}a^{7}+19\!\cdots\!26a^{6}+27\!\cdots\!93a^{5}-23\!\cdots\!29a^{4}-13\!\cdots\!68a^{3}-12\!\cdots\!17a^{2}+25\!\cdots\!10a+81\!\cdots\!39$, $\frac{14\!\cdots\!06}{11}a^{10}+\frac{88\!\cdots\!03}{11}a^{9}-\frac{36\!\cdots\!93}{11}a^{8}-\frac{85\!\cdots\!74}{11}a^{7}-73\!\cdots\!08a^{6}+25\!\cdots\!87a^{5}+41\!\cdots\!36a^{4}-22\!\cdots\!01a^{3}-17\!\cdots\!08a^{2}-19\!\cdots\!09a+30\!\cdots\!32$, $\frac{2340553722860}{11}a^{10}-2816409554162a^{9}+\frac{57299187322951}{11}a^{8}-1586336993107a^{7}-15930029104388a^{6}+46257242587945a^{5}-58975531501248a^{4}-17964296352530a^{3}+259619904373511a^{2}-610485394336235a+695406816483994$, $\frac{14\!\cdots\!43}{11}a^{10}+\frac{28\!\cdots\!31}{11}a^{9}+\frac{38\!\cdots\!14}{11}a^{8}+\frac{23\!\cdots\!20}{11}a^{7}-55\!\cdots\!19a^{6}-24\!\cdots\!97a^{5}-56\!\cdots\!37a^{4}-93\!\cdots\!12a^{3}-96\!\cdots\!60a^{2}+18\!\cdots\!39a+38\!\cdots\!05$, $\frac{24\!\cdots\!23}{11}a^{10}+\frac{11\!\cdots\!58}{11}a^{9}-\frac{46\!\cdots\!04}{11}a^{8}-\frac{18\!\cdots\!02}{11}a^{7}-37\!\cdots\!82a^{6}-59\!\cdots\!68a^{5}-56\!\cdots\!16a^{4}+25\!\cdots\!13a^{3}+27\!\cdots\!37a^{2}+75\!\cdots\!16a+14\!\cdots\!76$ (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: | \( 70243113861.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 70243113861.9 \cdot 2}{2\cdot\sqrt{2090387445666559894871442426539}}\cr\approx \mathstrut & 0.951523894587 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 110 |
The 11 conjugacy class representatives for $F_{11}$ |
Character table for $F_{11}$ |
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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.11.10.1 | $x^{11} + 7$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ |
\(11\) | 11.11.21.10 | $x^{11} + 484 x^{2} + 1210 x + 11$ | $11$ | $1$ | $21$ | $F_{11}$ | $[21/10]_{10}$ |