Normalized defining polynomial
\( x^{15} - 2 x^{14} - 3 x^{13} + 8 x^{12} + 4 x^{11} - 19 x^{10} + 3 x^{9} + 33 x^{8} - 23 x^{7} + \cdots - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1005792029461681407\) \(\medspace = -\,3^{6}\cdot 11^{8}\cdot 23^{5}\) | 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: | \(15.86\) | 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: | $3^{1/2}11^{4/5}23^{1/2}\approx 56.56381504121475$ | ||
Ramified primes: | \(3\), \(11\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{43332}a^{14}-\frac{2483}{43332}a^{13}+\frac{26}{157}a^{12}-\frac{720}{3611}a^{11}+\frac{79}{3611}a^{10}-\frac{12079}{43332}a^{9}-\frac{1649}{21666}a^{8}+\frac{7067}{43332}a^{7}+\frac{8105}{21666}a^{6}-\frac{19405}{43332}a^{5}+\frac{166}{3611}a^{4}+\frac{4061}{14444}a^{3}+\frac{438}{3611}a^{2}+\frac{2773}{43332}a-\frac{4477}{43332}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{9859}{10833}a^{14}-\frac{13859}{7222}a^{13}-\frac{1192}{471}a^{12}+\frac{28261}{3611}a^{11}+\frac{26341}{10833}a^{10}-\frac{66099}{3611}a^{9}+\frac{130531}{21666}a^{8}+\frac{659429}{21666}a^{7}-\frac{297350}{10833}a^{6}-\frac{143071}{3611}a^{5}+\frac{990823}{21666}a^{4}+\frac{224755}{7222}a^{3}-\frac{251716}{10833}a^{2}-\frac{54123}{3611}a-\frac{42701}{21666}$, $\frac{43373}{43332}a^{14}-\frac{101803}{43332}a^{13}-\frac{347}{157}a^{12}+\frac{31867}{3611}a^{11}+\frac{3239}{3611}a^{10}-\frac{841895}{43332}a^{9}+\frac{214049}{21666}a^{8}+\frac{1286383}{43332}a^{7}-\frac{729329}{21666}a^{6}-\frac{1532249}{43332}a^{5}+\frac{187356}{3611}a^{4}+\frac{354273}{14444}a^{3}-\frac{79539}{3611}a^{2}-\frac{622951}{43332}a-\frac{96893}{43332}$, $\frac{43373}{43332}a^{14}-\frac{101803}{43332}a^{13}-\frac{347}{157}a^{12}+\frac{31867}{3611}a^{11}+\frac{3239}{3611}a^{10}-\frac{841895}{43332}a^{9}+\frac{214049}{21666}a^{8}+\frac{1286383}{43332}a^{7}-\frac{729329}{21666}a^{6}-\frac{1532249}{43332}a^{5}+\frac{187356}{3611}a^{4}+\frac{354273}{14444}a^{3}-\frac{79539}{3611}a^{2}-\frac{622951}{43332}a-\frac{140225}{43332}$, $\frac{9607}{43332}a^{14}-\frac{9601}{14444}a^{13}-\frac{172}{471}a^{12}+\frac{8858}{3611}a^{11}-\frac{1682}{10833}a^{10}-\frac{76987}{14444}a^{9}+\frac{35867}{10833}a^{8}+\frac{330859}{43332}a^{7}-\frac{219529}{21666}a^{6}-\frac{123557}{14444}a^{5}+\frac{356911}{21666}a^{4}+\frac{109113}{14444}a^{3}-\frac{79900}{10833}a^{2}-\frac{113727}{14444}a-\frac{90193}{43332}$, $\frac{7975}{7222}a^{14}-\frac{20867}{7222}a^{13}-\frac{689}{471}a^{12}+\frac{106372}{10833}a^{11}-\frac{7789}{3611}a^{10}-\frac{427843}{21666}a^{9}+\frac{182011}{10833}a^{8}+\frac{186609}{7222}a^{7}-\frac{471616}{10833}a^{6}-\frac{167965}{7222}a^{5}+\frac{653902}{10833}a^{4}+\frac{121129}{21666}a^{3}-\frac{79386}{3611}a^{2}-\frac{163373}{21666}a-\frac{9965}{21666}$, $\frac{412}{10833}a^{14}+\frac{15889}{21666}a^{13}-\frac{327}{157}a^{12}-\frac{13678}{10833}a^{11}+\frac{80030}{10833}a^{10}-\frac{2604}{3611}a^{9}-\frac{337903}{21666}a^{8}+\frac{78997}{7222}a^{7}+\frac{240107}{10833}a^{6}-\frac{343151}{10833}a^{5}-\frac{171453}{7222}a^{4}+\frac{1007681}{21666}a^{3}+\frac{146923}{10833}a^{2}-\frac{76568}{3611}a-\frac{65737}{7222}$, $\frac{6057}{14444}a^{14}-\frac{46091}{43332}a^{13}-\frac{686}{471}a^{12}+\frac{17588}{3611}a^{11}+\frac{20270}{10833}a^{10}-\frac{516469}{43332}a^{9}+\frac{9044}{3611}a^{8}+\frac{895603}{43332}a^{7}-\frac{118723}{7222}a^{6}-\frac{1257955}{43332}a^{5}+\frac{704129}{21666}a^{4}+\frac{453021}{14444}a^{3}-\frac{219509}{10833}a^{2}-\frac{844685}{43332}a-\frac{56355}{14444}$ | 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: | \( 824.556928356 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 824.556928356 \cdot 1}{2\cdot\sqrt{1005792029461681407}}\cr\approx \mathstrut & 0.317852497985 \end{aligned}\]
Galois group
$C_5^2:D_6$ (as 15T18):
A solvable group of order 300 |
The 14 conjugacy class representatives for $((C_5^2 : C_3):C_2):C_2$ |
Character table for $((C_5^2 : C_3):C_2):C_2$ |
Intermediate fields
3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 sibling: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 siblings: | 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.3.0.1}{3} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.10.8.4 | $x^{10} - 165 x^{5} - 4356$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |