Normalized defining polynomial
\( x^{11} - 3x^{10} - 36x^{9} + 60x^{8} + 213x^{7} - 9x^{6} - 114x^{5} - 180x^{4} - 108x^{3} + 6x^{2} + 36x + 54 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(63019333158425674204677255696384\) \(\medspace = 2^{12}\cdot 3^{14}\cdot 7^{2}\cdot 23^{2}\cdot 137^{8}\) | 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: | \(777.79\) | 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: | $2^{43/24}3^{85/54}7^{1/2}23^{1/2}137^{4/5}\approx 12681.160459728126$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(23\), \(137\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{27000}a^{10}+\frac{41}{1125}a^{9}-\frac{23}{750}a^{8}-\frac{299}{1125}a^{7}-\frac{2833}{9000}a^{6}+\frac{157}{500}a^{5}-\frac{97}{1125}a^{4}-\frac{27}{250}a^{3}+\frac{2}{5}a^{2}-\frac{899}{4500}a-\frac{269}{1500}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | 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: | $\frac{58471}{6750}a^{10}-\frac{18649}{4500}a^{9}-\frac{461539}{1500}a^{8}-\frac{1090889}{4500}a^{7}+\frac{3308689}{4500}a^{6}+\frac{166019}{250}a^{5}+\frac{369002}{1125}a^{4}-\frac{33309}{125}a^{3}-\frac{3032}{5}a^{2}-\frac{552179}{1125}a-\frac{227023}{750}$, $\frac{19564411}{4500}a^{10}-\frac{14135567}{1500}a^{9}-\frac{61824937}{500}a^{8}+\frac{390119513}{1500}a^{7}+\frac{346841881}{750}a^{6}-\frac{49979522}{125}a^{5}+\frac{5401141}{375}a^{4}-\frac{71022041}{125}a^{3}+\frac{710712}{5}a^{2}-\frac{82832489}{750}a+\frac{25009533}{125}$, $\frac{15007510961}{13500}a^{10}-\frac{9184635667}{4500}a^{9}-\frac{63649347037}{1500}a^{8}+\frac{79030164913}{4500}a^{7}+\frac{289553895778}{1125}a^{6}+\frac{36023709251}{125}a^{5}+\frac{232986135191}{1125}a^{4}+\frac{4958192428}{125}a^{3}-\frac{371152016}{5}a^{2}-\frac{178478908939}{2250}a-\frac{19449889442}{375}$, $\frac{363317363}{3375}a^{10}-\frac{1880220619}{4500}a^{9}-\frac{5264387959}{1500}a^{8}+\frac{43036151941}{4500}a^{7}+\frac{65570640859}{4500}a^{6}-\frac{3515856661}{250}a^{5}-\frac{152874313}{1125}a^{4}-\frac{2330677204}{125}a^{3}+\frac{24095048}{5}a^{2}-\frac{4017895049}{1125}a+\frac{5041575287}{750}$, $\frac{100471139344}{3375}a^{10}-\frac{3258544892297}{4500}a^{9}+\frac{2088874621783}{1500}a^{8}+\frac{102830933823983}{4500}a^{7}-\frac{233241483936583}{4500}a^{6}-\frac{24228958187093}{250}a^{5}+\frac{138622023367306}{1125}a^{4}+\frac{1615369176973}{125}a^{3}+\frac{82063622664}{5}a^{2}+\frac{48002352941738}{1125}a-\frac{38891795645519}{750}$, $\frac{7774529}{2700}a^{10}-\frac{113563}{900}a^{9}-\frac{30371593}{300}a^{8}-\frac{108117143}{900}a^{7}+\frac{40791742}{225}a^{6}+\frac{6868139}{25}a^{5}+\frac{56920349}{225}a^{4}+\frac{2235417}{25}a^{3}-54806a^{2}-\frac{35069971}{450}a-\frac{4824338}{75}$, $\frac{14\!\cdots\!93}{6750}a^{10}-\frac{47\!\cdots\!67}{4500}a^{9}-\frac{87\!\cdots\!87}{1500}a^{8}+\frac{10\!\cdots\!13}{4500}a^{7}+\frac{90\!\cdots\!37}{4500}a^{6}-\frac{14\!\cdots\!73}{250}a^{5}-\frac{15\!\cdots\!84}{1125}a^{4}-\frac{15\!\cdots\!72}{125}a^{3}+\frac{13\!\cdots\!84}{5}a^{2}+\frac{89\!\cdots\!43}{1125}a+\frac{47\!\cdots\!41}{750}$, $\frac{14\!\cdots\!43}{3375}a^{10}-\frac{23\!\cdots\!96}{1125}a^{9}-\frac{90\!\cdots\!87}{750}a^{8}+\frac{51\!\cdots\!69}{1125}a^{7}+\frac{23\!\cdots\!87}{2250}a^{6}-\frac{79\!\cdots\!98}{125}a^{5}-\frac{30\!\cdots\!93}{1125}a^{4}-\frac{36\!\cdots\!94}{125}a^{3}-\frac{30\!\cdots\!97}{5}a^{2}-\frac{33\!\cdots\!64}{1125}a+\frac{36\!\cdots\!66}{375}$ (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: | \( 21261383857900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{2}\cdot 21261383857900 \cdot 1}{2\cdot\sqrt{63019333158425674204677255696384}}\cr\approx \mathstrut & 6.76697134460338 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 19958400 |
The 31 conjugacy class representatives for $A_{11}$ |
Character table for $A_{11}$ |
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 | R | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(137\) | $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |