Normalized defining polynomial
\( x^{15} - 7 x^{14} + 32 x^{13} - 114 x^{12} + 352 x^{11} - 874 x^{10} + 1982 x^{9} - 3764 x^{8} + \cdots - 500 \)
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: | \(-42630068813240240000000\) \(\medspace = -\,2^{10}\cdot 5^{7}\cdot 127^{7}\) | 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: | \(32.26\) | 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: | $2^{2/3}5^{1/2}127^{1/2}\approx 40.00124664765448$ | ||
Ramified primes: | \(2\), \(5\), \(127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-635}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{8}-\frac{1}{10}a^{6}-\frac{1}{2}a^{5}+\frac{2}{5}a^{4}-\frac{1}{2}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{10}a^{9}-\frac{1}{10}a^{7}-\frac{1}{10}a^{5}+\frac{1}{10}a^{3}$, $\frac{1}{20}a^{10}-\frac{1}{20}a^{9}-\frac{1}{5}a^{7}+\frac{3}{20}a^{6}+\frac{1}{20}a^{5}-\frac{1}{2}a^{4}+\frac{1}{5}a^{3}+\frac{3}{10}a^{2}$, $\frac{1}{20}a^{11}-\frac{1}{20}a^{9}-\frac{1}{20}a^{7}-\frac{9}{20}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{40}a^{12}-\frac{1}{40}a^{10}-\frac{1}{20}a^{9}+\frac{1}{40}a^{8}-\frac{1}{5}a^{7}+\frac{9}{40}a^{6}+\frac{1}{20}a^{5}-\frac{1}{20}a^{4}+\frac{1}{5}a^{3}+\frac{3}{10}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{363800}a^{13}-\frac{3113}{363800}a^{12}+\frac{201}{72760}a^{11}+\frac{5631}{363800}a^{10}-\frac{1759}{363800}a^{9}+\frac{2239}{72760}a^{8}+\frac{53867}{363800}a^{7}-\frac{72331}{363800}a^{6}+\frac{67079}{181900}a^{5}-\frac{49021}{181900}a^{4}-\frac{31213}{90950}a^{3}-\frac{8217}{18190}a^{2}-\frac{3878}{9095}a-\frac{455}{3638}$, $\frac{1}{9458800}a^{14}-\frac{1}{945880}a^{13}-\frac{9063}{1182350}a^{12}-\frac{93217}{4729400}a^{11}-\frac{2339}{2364700}a^{10}-\frac{104131}{4729400}a^{9}+\frac{51811}{4729400}a^{8}+\frac{147521}{945880}a^{7}+\frac{49323}{1891760}a^{6}-\frac{290161}{1182350}a^{5}+\frac{257611}{4729400}a^{4}+\frac{812561}{2364700}a^{3}+\frac{38745}{94588}a^{2}+\frac{214507}{472940}a+\frac{167}{884}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $5$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{11423}{9458800}a^{14}-\frac{22743}{4729400}a^{13}+\frac{9156}{591175}a^{12}-\frac{210611}{4729400}a^{11}+\frac{297319}{2364700}a^{10}-\frac{914221}{4729400}a^{9}+\frac{2375193}{4729400}a^{8}-\frac{3118341}{4729400}a^{7}+\frac{11771961}{9458800}a^{6}-\frac{827097}{591175}a^{5}+\frac{11679149}{4729400}a^{4}-\frac{5452639}{2364700}a^{3}+\frac{990383}{472940}a^{2}-\frac{1109643}{472940}a+\frac{121379}{94588}$, $\frac{2391}{378352}a^{14}-\frac{94051}{2364700}a^{13}+\frac{101542}{591175}a^{12}-\frac{68881}{118235}a^{11}+\frac{4114839}{2364700}a^{10}-\frac{9609841}{2364700}a^{9}+\frac{1707713}{189176}a^{8}-\frac{9493403}{591175}a^{7}+\frac{254059369}{9458800}a^{6}-\frac{43014479}{1182350}a^{5}+\frac{212072409}{4729400}a^{4}-\frac{6127889}{139100}a^{3}+\frac{16667923}{472940}a^{2}-\frac{10008381}{472940}a+\frac{390215}{94588}$, $\frac{52383}{9458800}a^{14}-\frac{9277}{236470}a^{13}+\frac{413337}{2364700}a^{12}-\frac{1407063}{2364700}a^{11}+\frac{4129313}{2364700}a^{10}-\frac{9611709}{2364700}a^{9}+\frac{40310163}{4729400}a^{8}-\frac{6911511}{472940}a^{7}+\frac{42301837}{1891760}a^{6}-\frac{65072591}{2364700}a^{5}+\frac{138003273}{4729400}a^{4}-\frac{56101667}{2364700}a^{3}+\frac{7247793}{472940}a^{2}-\frac{3622629}{472940}a+\frac{151791}{94588}$, $\frac{40091}{9458800}a^{14}-\frac{27247}{945880}a^{13}+\frac{143847}{1182350}a^{12}-\frac{1897837}{4729400}a^{11}+\frac{1368503}{1182350}a^{10}-\frac{12301301}{4729400}a^{9}+\frac{25142241}{4729400}a^{8}-\frac{1690795}{189176}a^{7}+\frac{1463309}{111280}a^{6}-\frac{9182218}{591175}a^{5}+\frac{4356673}{278200}a^{4}-\frac{27985889}{2364700}a^{3}+\frac{2394419}{472940}a^{2}-\frac{309823}{472940}a-\frac{8249}{94588}$, $\frac{27987}{4729400}a^{14}-\frac{8287}{118235}a^{13}+\frac{1851947}{4729400}a^{12}-\frac{1798257}{1182350}a^{11}+\frac{22514523}{4729400}a^{10}-\frac{7407003}{591175}a^{9}+\frac{129626889}{4729400}a^{8}-\frac{6104636}{118235}a^{7}+\frac{7730507}{94588}a^{6}-\frac{64623427}{591175}a^{5}+\frac{137687911}{1182350}a^{4}-\frac{56628399}{591175}a^{3}+\frac{12240633}{236470}a^{2}-\frac{1434453}{118235}a-\frac{16634}{23647}$, $\frac{61547}{9458800}a^{14}-\frac{193907}{4729400}a^{13}+\frac{842217}{4729400}a^{12}-\frac{2883539}{4729400}a^{11}+\frac{8630047}{4729400}a^{10}-\frac{20174029}{4729400}a^{9}+\frac{22396181}{2364700}a^{8}-\frac{79683809}{4729400}a^{7}+\frac{264252819}{9458800}a^{6}-\frac{22486603}{591175}a^{5}+\frac{218172931}{4729400}a^{4}-\frac{107845511}{2364700}a^{3}+\frac{17323951}{472940}a^{2}-\frac{10240897}{472940}a+\frac{613869}{94588}$, $\frac{3603}{556400}a^{14}-\frac{95323}{2364700}a^{13}+\frac{103277}{591175}a^{12}-\frac{1411841}{2364700}a^{11}+\frac{2114129}{1182350}a^{10}-\frac{9839041}{2364700}a^{9}+\frac{43719321}{4729400}a^{8}-\frac{38610801}{2364700}a^{7}+\frac{255683897}{9458800}a^{6}-\frac{21355819}{591175}a^{5}+\frac{207560793}{4729400}a^{4}-\frac{98218113}{2364700}a^{3}+\frac{15794303}{472940}a^{2}-\frac{8665601}{472940}a+\frac{325095}{94588}$ (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: | \( 1804580.53513 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1804580.53513 \cdot 1}{2\cdot\sqrt{42630068813240240000000}}\cr\approx \mathstrut & 3.37891549376 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.2540.1, 5.1.403225.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 30 |
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 | $15$ | R | $15$ | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(127\) | $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |