Normalized defining polynomial
\( x^{11} - 55x^{9} + 1100x^{7} - 9625x^{5} + 34375x^{3} - 34375x - 12675 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[11, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(448727830698946884765625\) \(\medspace = 5^{10}\cdot 11^{16}\) | 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: | \(141.31\) | 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: | $5^{10/11}11^{84/55}\approx 168.23560841235079$ | ||
Ramified primes: | \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{355}a^{6}-\frac{25}{71}a^{5}-\frac{6}{71}a^{4}-\frac{14}{71}a^{3}-\frac{26}{71}a^{2}-\frac{1}{71}a+\frac{21}{71}$, $\frac{1}{355}a^{7}-\frac{7}{71}a^{5}+\frac{17}{71}a^{4}-\frac{1}{71}a^{3}+\frac{15}{71}a^{2}-\frac{33}{71}a-\frac{2}{71}$, $\frac{1}{355}a^{8}-\frac{6}{71}a^{5}+\frac{2}{71}a^{4}+\frac{22}{71}a^{3}-\frac{20}{71}a^{2}+\frac{34}{71}a+\frac{25}{71}$, $\frac{1}{355}a^{9}+\frac{33}{71}a^{5}-\frac{16}{71}a^{4}-\frac{14}{71}a^{3}+\frac{35}{71}a^{2}-\frac{5}{71}a-\frac{9}{71}$, $\frac{1}{355}a^{10}-\frac{9}{71}a^{5}-\frac{18}{71}a^{4}+\frac{2}{71}a^{3}+\frac{25}{71}a^{2}+\frac{14}{71}a+\frac{14}{71}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $\frac{2}{355}a^{9}+\frac{3}{355}a^{8}-\frac{83}{355}a^{7}-\frac{114}{355}a^{6}+3a^{5}+\frac{241}{71}a^{4}-\frac{839}{71}a^{3}-\frac{543}{71}a^{2}+\frac{531}{71}a+\frac{314}{71}$, $\frac{1}{355}a^{10}-\frac{3}{355}a^{9}-\frac{43}{355}a^{8}+\frac{132}{355}a^{7}+\frac{598}{355}a^{6}-\frac{388}{71}a^{5}-\frac{548}{71}a^{4}+\frac{2025}{71}a^{3}+\frac{276}{71}a^{2}-\frac{1914}{71}a-\frac{739}{71}$, $\frac{1}{355}a^{10}-\frac{2}{355}a^{9}-\frac{10}{71}a^{8}+\frac{98}{355}a^{7}+\frac{878}{355}a^{6}-\frac{330}{71}a^{5}-\frac{1274}{71}a^{4}+\frac{2160}{71}a^{3}+\frac{3453}{71}a^{2}-\frac{4368}{71}a-\frac{2501}{71}$, $\frac{1}{355}a^{10}-\frac{1}{355}a^{9}-\frac{43}{355}a^{8}+\frac{26}{355}a^{7}+\frac{624}{355}a^{6}-\frac{17}{71}a^{5}-\frac{692}{71}a^{4}-\frac{249}{71}a^{3}+\frac{991}{71}a^{2}+\frac{909}{71}a+\frac{214}{71}$, $\frac{2}{355}a^{10}+\frac{1}{71}a^{9}-\frac{82}{355}a^{8}-\frac{187}{355}a^{7}+\frac{1064}{355}a^{6}+\frac{411}{71}a^{5}-\frac{968}{71}a^{4}-\frac{1314}{71}a^{3}+\frac{1500}{71}a^{2}+\frac{973}{71}a+\frac{61}{71}$, $\frac{1}{355}a^{10}-\frac{2}{355}a^{9}-\frac{44}{355}a^{8}+\frac{96}{355}a^{7}+\frac{643}{355}a^{6}-\frac{299}{71}a^{5}-\frac{667}{71}a^{4}+\frac{1608}{71}a^{3}+\frac{751}{71}a^{2}-\frac{1591}{71}a-\frac{821}{71}$, $\frac{3}{355}a^{10}-\frac{7}{355}a^{9}-\frac{129}{355}a^{8}+\frac{266}{355}a^{7}+\frac{1947}{355}a^{6}-\frac{676}{71}a^{5}-\frac{2532}{71}a^{4}+\frac{3538}{71}a^{3}+\frac{6543}{71}a^{2}-\frac{6656}{71}a-\frac{4016}{71}$, $\frac{2}{355}a^{9}+\frac{12}{355}a^{8}-\frac{54}{355}a^{7}-\frac{388}{355}a^{6}+\frac{61}{71}a^{5}+\frac{763}{71}a^{4}+\frac{255}{71}a^{3}-\frac{2110}{71}a^{2}-\frac{1266}{71}a+\frac{194}{71}$, $\frac{2}{355}a^{9}-\frac{3}{355}a^{8}-\frac{19}{71}a^{7}+\frac{143}{355}a^{6}+\frac{298}{71}a^{5}-\frac{452}{71}a^{4}-\frac{1575}{71}a^{3}+\frac{2371}{71}a^{2}+\frac{537}{71}a-\frac{379}{71}$, $\frac{4}{355}a^{10}-\frac{12}{355}a^{9}-\frac{212}{355}a^{8}+\frac{504}{355}a^{7}+\frac{762}{71}a^{6}-\frac{1307}{71}a^{5}-\frac{5082}{71}a^{4}+\frac{4929}{71}a^{3}+\frac{8341}{71}a^{2}+\frac{582}{71}a-\frac{826}{71}$ (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: | \( 677575079.592 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 677575079.592 \cdot 1}{2\cdot\sqrt{448727830698946884765625}}\cr\approx \mathstrut & 1.03577608282 \end{aligned}\] (assuming GRH)
Galois group
$C_{11}:C_5$ (as 11T3):
A solvable group of order 55 |
The 7 conjugacy class representatives for $C_{11}:C_5$ |
Character table for $C_{11}:C_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.11.10.1 | $x^{11} + 5$ | $11$ | $1$ | $10$ | $C_{11}:C_5$ | $[\ ]_{11}^{5}$ |
\(11\) | 11.11.16.4 | $x^{11} + 22 x^{6} + 11$ | $11$ | $1$ | $16$ | $C_{11}:C_5$ | $[8/5]_{5}$ |