Normalized defining polynomial
\( x^{11} - 3x^{10} - 6x^{9} + 12x^{8} + 18x^{7} - 54x^{6} - 96x^{5} + 72x^{4} + 126x^{3} - 206x^{2} - 336x - 96 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(22674816000000000\) \(\medspace = 2^{16}\cdot 3^{11}\cdot 5^{9}\) | 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: | \(30.68\) | 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: | $2^{15/8}3^{43/36}5^{71/60}\approx 91.50410531097494$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{19503024}a^{10}-\frac{37205}{6501008}a^{9}+\frac{821023}{9751512}a^{8}-\frac{176801}{1625252}a^{7}-\frac{928919}{9751512}a^{6}+\frac{3682321}{9751512}a^{5}-\frac{184061}{2437878}a^{4}+\frac{231061}{2437878}a^{3}+\frac{1508061}{3250504}a^{2}+\frac{977263}{3250504}a-\frac{91447}{812626}$
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)
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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: | $\frac{164545}{9751512}a^{10}-\frac{342575}{9751512}a^{9}-\frac{197487}{1625252}a^{8}+\frac{58139}{2437878}a^{7}+\frac{1097989}{4875756}a^{6}-\frac{354641}{1625252}a^{5}-\frac{965164}{1218939}a^{4}-\frac{400417}{1218939}a^{3}+\frac{421885}{1625252}a^{2}-\frac{317797}{1625252}a-\frac{157286}{406313}$, $\frac{156563}{812626}a^{10}-\frac{1834475}{2437878}a^{9}-\frac{169357}{406313}a^{8}+\frac{2971054}{1218939}a^{7}+\frac{517197}{406313}a^{6}-\frac{13379939}{1218939}a^{5}-\frac{10185275}{1218939}a^{4}+\frac{23152879}{1218939}a^{3}+\frac{7821793}{1218939}a^{2}-\frac{16742161}{406313}a-\frac{11368837}{406313}$, $\frac{1565317}{19503024}a^{10}-\frac{1589321}{6501008}a^{9}-\frac{1169735}{3250504}a^{8}+\frac{3025189}{4875756}a^{7}+\frac{8254013}{9751512}a^{6}-\frac{31318835}{9751512}a^{5}-\frac{4505519}{812626}a^{4}+\frac{5819639}{2437878}a^{3}+\frac{21263851}{9751512}a^{2}-\frac{45308133}{3250504}a-\frac{10037937}{812626}$, $\frac{409349}{19503024}a^{10}-\frac{404411}{19503024}a^{9}-\frac{4167557}{9751512}a^{8}+\frac{3978041}{4875756}a^{7}+\frac{8146709}{9751512}a^{6}-\frac{37556435}{9751512}a^{5}-\frac{584691}{812626}a^{4}+\frac{5079513}{812626}a^{3}-\frac{5956383}{3250504}a^{2}-\frac{27150029}{3250504}a-\frac{1746565}{812626}$, $\frac{875047}{6501008}a^{10}-\frac{3727721}{6501008}a^{9}-\frac{796525}{9751512}a^{8}+\frac{8107373}{4875756}a^{7}+\frac{4251245}{9751512}a^{6}-\frac{75015355}{9751512}a^{5}-\frac{7596887}{2437878}a^{4}+\frac{32597087}{2437878}a^{3}-\frac{573103}{3250504}a^{2}-\frac{81874509}{3250504}a-\frac{7485497}{812626}$, $\frac{45343}{9751512}a^{10}+\frac{25261}{3250504}a^{9}-\frac{376423}{4875756}a^{8}-\frac{132253}{812626}a^{7}+\frac{1681867}{4875756}a^{6}+\frac{1239297}{1625252}a^{5}-\frac{1021529}{1218939}a^{4}-\frac{3438599}{1218939}a^{3}+\frac{5598085}{4875756}a^{2}+\frac{12542445}{1625252}a+\frac{2380722}{406313}$ (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: | \( 108767.621652 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 108767.621652 \cdot 1}{2\cdot\sqrt{22674816000000000}}\cr\approx \mathstrut & 4.50305458270 \end{aligned}\] (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 | R | R | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 | $[\ ]$ |
2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.8.14.6 | $x^{8} + 2 x^{7} + 2 x^{4} + 6$ | $8$ | $1$ | $14$ | $C_2^3:C_7$ | $[2, 2, 2]^{7}$ | |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.9.10.1 | $x^{9} + 3 x^{2} + 3$ | $9$ | $1$ | $10$ | $C_3^2:Q_8$ | $[5/4, 5/4]_{4}^{2}$ | |
\(5\) | 5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
5.6.4.2 | $x^{6} + 10 x^{3} - 25$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |