Normalized defining polynomial
\( x^{15} - x^{14} + 17 x^{13} + 103 x^{12} - 709 x^{11} - 855 x^{10} - 10199 x^{9} - 201819 x^{8} + \cdots - 19034128 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-612284023294691059283807682781787\) \(\medspace = -\,41^{12}\cdot 83^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(153.39\) | 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: | $41^{4/5}83^{1/2}\approx 177.7322060855116$ | ||
Ramified primes: | \(41\), \(83\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-83}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{20}a^{9}+\frac{1}{20}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{1}{10}a^{4}+\frac{1}{4}a^{3}+\frac{3}{20}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{20}a^{10}+\frac{1}{20}a^{8}-\frac{1}{5}a^{7}-\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{3}{20}a^{4}-\frac{1}{10}a^{3}+\frac{3}{20}a^{2}+\frac{1}{5}$, $\frac{1}{20}a^{11}-\frac{1}{5}a^{7}-\frac{1}{10}a^{6}-\frac{3}{20}a^{5}-\frac{1}{5}a^{4}-\frac{1}{10}a^{3}+\frac{1}{10}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5200}a^{12}-\frac{43}{2600}a^{11}+\frac{21}{1300}a^{10}+\frac{11}{1040}a^{9}+\frac{3}{65}a^{8}-\frac{41}{1300}a^{7}+\frac{1011}{5200}a^{6}+\frac{469}{2600}a^{5}-\frac{6}{325}a^{4}-\frac{183}{400}a^{3}-\frac{23}{1300}a^{2}+\frac{317}{650}a-\frac{31}{650}$, $\frac{1}{5200}a^{13}-\frac{2}{325}a^{11}-\frac{1}{5200}a^{10}+\frac{3}{520}a^{9}+\frac{57}{650}a^{8}+\frac{427}{5200}a^{7}-\frac{259}{1300}a^{6}+\frac{253}{1300}a^{5}-\frac{51}{208}a^{4}-\frac{293}{2600}a^{3}-\frac{369}{1300}a^{2}-\frac{69}{650}a-\frac{98}{325}$, $\frac{1}{91\!\cdots\!00}a^{14}-\frac{67\!\cdots\!19}{91\!\cdots\!00}a^{13}-\frac{13\!\cdots\!01}{91\!\cdots\!00}a^{12}-\frac{13\!\cdots\!99}{91\!\cdots\!00}a^{11}+\frac{51\!\cdots\!13}{91\!\cdots\!00}a^{10}-\frac{16\!\cdots\!09}{91\!\cdots\!00}a^{9}+\frac{28\!\cdots\!63}{91\!\cdots\!00}a^{8}+\frac{77\!\cdots\!87}{91\!\cdots\!00}a^{7}+\frac{64\!\cdots\!97}{91\!\cdots\!00}a^{6}+\frac{53\!\cdots\!91}{18\!\cdots\!00}a^{5}-\frac{60\!\cdots\!97}{91\!\cdots\!00}a^{4}+\frac{44\!\cdots\!69}{91\!\cdots\!00}a^{3}-\frac{14\!\cdots\!01}{35\!\cdots\!00}a^{2}-\frac{52\!\cdots\!13}{16\!\cdots\!00}a-\frac{33\!\cdots\!27}{11\!\cdots\!00}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{5}\times C_{5}$, which has order $25$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{81\!\cdots\!91}{26\!\cdots\!50}a^{14}-\frac{28\!\cdots\!61}{10\!\cdots\!00}a^{13}+\frac{51\!\cdots\!51}{10\!\cdots\!00}a^{12}+\frac{89\!\cdots\!91}{26\!\cdots\!50}a^{11}-\frac{23\!\cdots\!13}{10\!\cdots\!00}a^{10}-\frac{31\!\cdots\!51}{10\!\cdots\!00}a^{9}-\frac{76\!\cdots\!97}{26\!\cdots\!50}a^{8}-\frac{66\!\cdots\!07}{10\!\cdots\!00}a^{7}-\frac{20\!\cdots\!07}{10\!\cdots\!00}a^{6}-\frac{74\!\cdots\!39}{21\!\cdots\!66}a^{5}-\frac{24\!\cdots\!23}{10\!\cdots\!00}a^{4}-\frac{57\!\cdots\!49}{10\!\cdots\!00}a^{3}-\frac{19\!\cdots\!06}{40\!\cdots\!55}a^{2}-\frac{10\!\cdots\!19}{37\!\cdots\!50}a-\frac{13\!\cdots\!73}{13\!\cdots\!75}$, $\frac{19\!\cdots\!41}{45\!\cdots\!00}a^{14}-\frac{86\!\cdots\!19}{45\!\cdots\!00}a^{13}-\frac{25\!\cdots\!07}{57\!\cdots\!50}a^{12}+\frac{89\!\cdots\!11}{45\!\cdots\!00}a^{11}-\frac{15\!\cdots\!87}{45\!\cdots\!00}a^{10}-\frac{10\!\cdots\!97}{22\!\cdots\!00}a^{9}-\frac{47\!\cdots\!57}{45\!\cdots\!00}a^{8}-\frac{10\!\cdots\!53}{45\!\cdots\!00}a^{7}+\frac{13\!\cdots\!69}{57\!\cdots\!50}a^{6}+\frac{13\!\cdots\!61}{91\!\cdots\!00}a^{5}+\frac{90\!\cdots\!63}{45\!\cdots\!00}a^{4}+\frac{16\!\cdots\!77}{22\!\cdots\!00}a^{3}+\frac{12\!\cdots\!04}{43\!\cdots\!75}a^{2}-\frac{36\!\cdots\!53}{80\!\cdots\!50}a+\frac{16\!\cdots\!74}{28\!\cdots\!75}$, $\frac{15\!\cdots\!99}{45\!\cdots\!00}a^{14}-\frac{65\!\cdots\!19}{11\!\cdots\!00}a^{13}+\frac{21\!\cdots\!97}{57\!\cdots\!50}a^{12}+\frac{15\!\cdots\!89}{45\!\cdots\!00}a^{11}-\frac{67\!\cdots\!09}{22\!\cdots\!00}a^{10}-\frac{43\!\cdots\!79}{11\!\cdots\!00}a^{9}-\frac{63\!\cdots\!83}{45\!\cdots\!00}a^{8}-\frac{36\!\cdots\!19}{57\!\cdots\!50}a^{7}-\frac{17\!\cdots\!63}{11\!\cdots\!00}a^{6}+\frac{16\!\cdots\!31}{91\!\cdots\!00}a^{5}-\frac{23\!\cdots\!89}{22\!\cdots\!00}a^{4}-\frac{21\!\cdots\!53}{57\!\cdots\!50}a^{3}+\frac{30\!\cdots\!87}{43\!\cdots\!75}a^{2}+\frac{18\!\cdots\!93}{80\!\cdots\!50}a+\frac{19\!\cdots\!86}{28\!\cdots\!75}$, $\frac{16\!\cdots\!47}{91\!\cdots\!00}a^{14}-\frac{62\!\cdots\!13}{91\!\cdots\!00}a^{13}+\frac{38\!\cdots\!33}{91\!\cdots\!00}a^{12}+\frac{12\!\cdots\!07}{91\!\cdots\!00}a^{11}-\frac{19\!\cdots\!49}{91\!\cdots\!00}a^{10}+\frac{56\!\cdots\!77}{91\!\cdots\!00}a^{9}-\frac{31\!\cdots\!59}{91\!\cdots\!00}a^{8}-\frac{27\!\cdots\!71}{91\!\cdots\!00}a^{7}+\frac{47\!\cdots\!59}{91\!\cdots\!00}a^{6}-\frac{42\!\cdots\!03}{18\!\cdots\!00}a^{5}-\frac{64\!\cdots\!39}{91\!\cdots\!00}a^{4}+\frac{18\!\cdots\!43}{91\!\cdots\!00}a^{3}-\frac{16\!\cdots\!41}{35\!\cdots\!00}a^{2}+\frac{15\!\cdots\!29}{16\!\cdots\!00}a-\frac{10\!\cdots\!29}{11\!\cdots\!00}$, $\frac{13\!\cdots\!57}{91\!\cdots\!00}a^{14}-\frac{51\!\cdots\!73}{91\!\cdots\!00}a^{13}+\frac{39\!\cdots\!23}{91\!\cdots\!00}a^{12}-\frac{17\!\cdots\!43}{91\!\cdots\!00}a^{11}-\frac{70\!\cdots\!49}{91\!\cdots\!00}a^{10}+\frac{28\!\cdots\!87}{91\!\cdots\!00}a^{9}-\frac{16\!\cdots\!49}{91\!\cdots\!00}a^{8}-\frac{19\!\cdots\!91}{91\!\cdots\!00}a^{7}-\frac{30\!\cdots\!51}{91\!\cdots\!00}a^{6}-\frac{17\!\cdots\!21}{18\!\cdots\!00}a^{5}-\frac{41\!\cdots\!59}{91\!\cdots\!00}a^{4}-\frac{69\!\cdots\!47}{91\!\cdots\!00}a^{3}-\frac{46\!\cdots\!69}{35\!\cdots\!00}a^{2}+\frac{19\!\cdots\!29}{16\!\cdots\!00}a-\frac{32\!\cdots\!79}{11\!\cdots\!00}$, $\frac{44\!\cdots\!53}{17\!\cdots\!00}a^{14}-\frac{26\!\cdots\!37}{17\!\cdots\!00}a^{13}+\frac{19\!\cdots\!07}{17\!\cdots\!00}a^{12}-\frac{19\!\cdots\!47}{17\!\cdots\!00}a^{11}-\frac{31\!\cdots\!41}{17\!\cdots\!00}a^{10}+\frac{16\!\cdots\!73}{17\!\cdots\!00}a^{9}-\frac{10\!\cdots\!91}{17\!\cdots\!00}a^{8}-\frac{56\!\cdots\!39}{17\!\cdots\!00}a^{7}+\frac{17\!\cdots\!81}{17\!\cdots\!00}a^{6}-\frac{13\!\cdots\!93}{35\!\cdots\!00}a^{5}-\frac{35\!\cdots\!51}{17\!\cdots\!00}a^{4}+\frac{35\!\cdots\!47}{17\!\cdots\!00}a^{3}-\frac{24\!\cdots\!97}{17\!\cdots\!00}a^{2}+\frac{85\!\cdots\!01}{30\!\cdots\!25}a-\frac{45\!\cdots\!97}{16\!\cdots\!75}$, $\frac{11\!\cdots\!13}{91\!\cdots\!00}a^{14}+\frac{20\!\cdots\!83}{91\!\cdots\!00}a^{13}+\frac{17\!\cdots\!57}{91\!\cdots\!00}a^{12}+\frac{14\!\cdots\!33}{91\!\cdots\!00}a^{11}-\frac{52\!\cdots\!81}{91\!\cdots\!00}a^{10}-\frac{36\!\cdots\!67}{91\!\cdots\!00}a^{9}-\frac{18\!\cdots\!01}{91\!\cdots\!00}a^{8}-\frac{25\!\cdots\!39}{91\!\cdots\!00}a^{7}-\frac{12\!\cdots\!49}{91\!\cdots\!00}a^{6}-\frac{58\!\cdots\!93}{18\!\cdots\!00}a^{5}-\frac{64\!\cdots\!31}{91\!\cdots\!00}a^{4}-\frac{12\!\cdots\!13}{91\!\cdots\!00}a^{3}+\frac{13\!\cdots\!67}{54\!\cdots\!60}a^{2}-\frac{50\!\cdots\!99}{16\!\cdots\!00}a+\frac{26\!\cdots\!99}{11\!\cdots\!00}$ (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: | \( 12620048295.521133 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 12620048295.521133 \cdot 25}{2\cdot\sqrt{612284023294691059283807682781787}}\cr\approx \mathstrut & 4.92928172727402 \end{aligned}\] (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
3.1.83.1, 5.1.19466667529.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{7}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $15$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | $15$ | $15$ | $15$ | R | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(41\) | 41.5.4.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
41.5.4.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
41.5.4.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.83.2t1.a.a | $1$ | $ 83 $ | \(\Q(\sqrt{-83}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.83.3t2.a.a | $2$ | $ 83 $ | 3.1.83.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.139523.5t2.a.b | $2$ | $ 41^{2} \cdot 83 $ | 5.1.19466667529.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.139523.5t2.a.a | $2$ | $ 41^{2} \cdot 83 $ | 5.1.19466667529.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.139523.15t2.a.b | $2$ | $ 41^{2} \cdot 83 $ | 15.1.612284023294691059283807682781787.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.139523.15t2.a.d | $2$ | $ 41^{2} \cdot 83 $ | 15.1.612284023294691059283807682781787.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.139523.15t2.a.a | $2$ | $ 41^{2} \cdot 83 $ | 15.1.612284023294691059283807682781787.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.139523.15t2.a.c | $2$ | $ 41^{2} \cdot 83 $ | 15.1.612284023294691059283807682781787.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |