Normalized defining polynomial
\( x^{15} - 4 x^{14} - 12 x^{13} + 63 x^{12} + 75 x^{11} - 365 x^{10} - 859 x^{9} - 1340 x^{8} - 2526 x^{7} + 965 x^{6} - 1651 x^{5} + 7887 x^{4} + 935 x^{3} + 418 x^{2} - 2167 x - 671 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-7789948783671344586860223\) \(\medspace = -\,3^{6}\cdot 11^{12}\cdot 23^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.65\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{47\!\cdots\!74}a^{14}+\frac{68\!\cdots\!51}{47\!\cdots\!74}a^{13}+\frac{27\!\cdots\!97}{23\!\cdots\!37}a^{12}+\frac{98\!\cdots\!49}{47\!\cdots\!74}a^{11}+\frac{21\!\cdots\!14}{23\!\cdots\!37}a^{10}+\frac{25\!\cdots\!20}{23\!\cdots\!37}a^{9}-\frac{13\!\cdots\!63}{47\!\cdots\!74}a^{8}+\frac{16\!\cdots\!73}{47\!\cdots\!74}a^{7}+\frac{11\!\cdots\!89}{23\!\cdots\!37}a^{6}+\frac{20\!\cdots\!99}{47\!\cdots\!74}a^{5}-\frac{21\!\cdots\!49}{47\!\cdots\!74}a^{4}+\frac{20\!\cdots\!17}{47\!\cdots\!74}a^{3}-\frac{47\!\cdots\!14}{23\!\cdots\!37}a^{2}+\frac{49\!\cdots\!61}{47\!\cdots\!74}a-\frac{21\!\cdots\!55}{47\!\cdots\!74}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{40\!\cdots\!43}{18\!\cdots\!37}a^{14}-\frac{22\!\cdots\!39}{18\!\cdots\!37}a^{13}-\frac{48\!\cdots\!21}{36\!\cdots\!74}a^{12}+\frac{63\!\cdots\!31}{36\!\cdots\!74}a^{11}-\frac{66\!\cdots\!58}{18\!\cdots\!37}a^{10}-\frac{17\!\cdots\!80}{18\!\cdots\!37}a^{9}-\frac{28\!\cdots\!81}{36\!\cdots\!74}a^{8}-\frac{14\!\cdots\!15}{18\!\cdots\!37}a^{7}-\frac{69\!\cdots\!67}{36\!\cdots\!74}a^{6}+\frac{17\!\cdots\!18}{18\!\cdots\!37}a^{5}-\frac{17\!\cdots\!97}{18\!\cdots\!37}a^{4}+\frac{43\!\cdots\!37}{18\!\cdots\!37}a^{3}-\frac{97\!\cdots\!23}{36\!\cdots\!74}a^{2}+\frac{12\!\cdots\!29}{36\!\cdots\!74}a-\frac{70\!\cdots\!45}{18\!\cdots\!37}$, $\frac{71\!\cdots\!77}{23\!\cdots\!37}a^{14}-\frac{34\!\cdots\!94}{23\!\cdots\!37}a^{13}-\frac{65\!\cdots\!17}{23\!\cdots\!37}a^{12}+\frac{10\!\cdots\!09}{47\!\cdots\!74}a^{11}+\frac{52\!\cdots\!15}{47\!\cdots\!74}a^{10}-\frac{61\!\cdots\!31}{47\!\cdots\!74}a^{9}-\frac{10\!\cdots\!89}{47\!\cdots\!74}a^{8}-\frac{43\!\cdots\!15}{23\!\cdots\!37}a^{7}-\frac{19\!\cdots\!21}{47\!\cdots\!74}a^{6}+\frac{64\!\cdots\!87}{47\!\cdots\!74}a^{5}-\frac{52\!\cdots\!11}{47\!\cdots\!74}a^{4}+\frac{74\!\cdots\!03}{47\!\cdots\!74}a^{3}-\frac{14\!\cdots\!67}{47\!\cdots\!74}a^{2}+\frac{42\!\cdots\!52}{23\!\cdots\!37}a+\frac{91\!\cdots\!16}{23\!\cdots\!37}$, $\frac{23\!\cdots\!53}{47\!\cdots\!74}a^{14}-\frac{73\!\cdots\!83}{47\!\cdots\!74}a^{13}-\frac{37\!\cdots\!55}{47\!\cdots\!74}a^{12}+\frac{64\!\cdots\!24}{23\!\cdots\!37}a^{11}+\frac{15\!\cdots\!54}{23\!\cdots\!37}a^{10}-\frac{39\!\cdots\!02}{23\!\cdots\!37}a^{9}-\frac{13\!\cdots\!68}{23\!\cdots\!37}a^{8}-\frac{43\!\cdots\!93}{47\!\cdots\!74}a^{7}-\frac{98\!\cdots\!41}{47\!\cdots\!74}a^{6}-\frac{82\!\cdots\!11}{47\!\cdots\!74}a^{5}-\frac{47\!\cdots\!47}{47\!\cdots\!74}a^{4}+\frac{87\!\cdots\!01}{47\!\cdots\!74}a^{3}+\frac{57\!\cdots\!57}{47\!\cdots\!74}a^{2}+\frac{84\!\cdots\!76}{23\!\cdots\!37}a+\frac{34\!\cdots\!45}{47\!\cdots\!74}$, $\frac{20\!\cdots\!59}{47\!\cdots\!74}a^{14}-\frac{44\!\cdots\!63}{23\!\cdots\!37}a^{13}-\frac{22\!\cdots\!33}{47\!\cdots\!74}a^{12}+\frac{14\!\cdots\!01}{47\!\cdots\!74}a^{11}+\frac{11\!\cdots\!69}{47\!\cdots\!74}a^{10}-\frac{44\!\cdots\!35}{23\!\cdots\!37}a^{9}-\frac{73\!\cdots\!00}{23\!\cdots\!37}a^{8}-\frac{13\!\cdots\!11}{47\!\cdots\!74}a^{7}-\frac{18\!\cdots\!38}{23\!\cdots\!37}a^{6}+\frac{16\!\cdots\!81}{23\!\cdots\!37}a^{5}-\frac{84\!\cdots\!44}{23\!\cdots\!37}a^{4}+\frac{14\!\cdots\!99}{47\!\cdots\!74}a^{3}-\frac{21\!\cdots\!38}{23\!\cdots\!37}a^{2}-\frac{27\!\cdots\!06}{23\!\cdots\!37}a+\frac{70\!\cdots\!73}{47\!\cdots\!74}$, $\frac{20\!\cdots\!59}{47\!\cdots\!74}a^{14}-\frac{44\!\cdots\!63}{23\!\cdots\!37}a^{13}-\frac{22\!\cdots\!33}{47\!\cdots\!74}a^{12}+\frac{14\!\cdots\!01}{47\!\cdots\!74}a^{11}+\frac{11\!\cdots\!69}{47\!\cdots\!74}a^{10}-\frac{44\!\cdots\!35}{23\!\cdots\!37}a^{9}-\frac{73\!\cdots\!00}{23\!\cdots\!37}a^{8}-\frac{13\!\cdots\!11}{47\!\cdots\!74}a^{7}-\frac{18\!\cdots\!38}{23\!\cdots\!37}a^{6}+\frac{16\!\cdots\!81}{23\!\cdots\!37}a^{5}-\frac{84\!\cdots\!44}{23\!\cdots\!37}a^{4}+\frac{14\!\cdots\!99}{47\!\cdots\!74}a^{3}-\frac{21\!\cdots\!38}{23\!\cdots\!37}a^{2}-\frac{27\!\cdots\!06}{23\!\cdots\!37}a-\frac{40\!\cdots\!01}{47\!\cdots\!74}$, $\frac{19\!\cdots\!53}{47\!\cdots\!74}a^{14}-\frac{16\!\cdots\!32}{23\!\cdots\!37}a^{13}-\frac{20\!\cdots\!75}{23\!\cdots\!37}a^{12}+\frac{33\!\cdots\!88}{23\!\cdots\!37}a^{11}+\frac{21\!\cdots\!34}{23\!\cdots\!37}a^{10}-\frac{34\!\cdots\!35}{47\!\cdots\!74}a^{9}-\frac{16\!\cdots\!17}{23\!\cdots\!37}a^{8}-\frac{32\!\cdots\!15}{23\!\cdots\!37}a^{7}-\frac{12\!\cdots\!99}{47\!\cdots\!74}a^{6}-\frac{16\!\cdots\!79}{47\!\cdots\!74}a^{5}-\frac{17\!\cdots\!19}{47\!\cdots\!74}a^{4}-\frac{91\!\cdots\!88}{23\!\cdots\!37}a^{3}+\frac{59\!\cdots\!90}{23\!\cdots\!37}a^{2}+\frac{30\!\cdots\!19}{23\!\cdots\!37}a+\frac{91\!\cdots\!73}{47\!\cdots\!74}$, $\frac{61\!\cdots\!25}{47\!\cdots\!74}a^{14}-\frac{42\!\cdots\!52}{23\!\cdots\!37}a^{13}-\frac{12\!\cdots\!21}{47\!\cdots\!74}a^{12}+\frac{72\!\cdots\!10}{23\!\cdots\!37}a^{11}+\frac{67\!\cdots\!82}{23\!\cdots\!37}a^{10}-\frac{34\!\cdots\!29}{47\!\cdots\!74}a^{9}-\frac{10\!\cdots\!09}{47\!\cdots\!74}a^{8}-\frac{24\!\cdots\!21}{47\!\cdots\!74}a^{7}-\frac{49\!\cdots\!05}{47\!\cdots\!74}a^{6}-\frac{64\!\cdots\!07}{47\!\cdots\!74}a^{5}-\frac{48\!\cdots\!03}{47\!\cdots\!74}a^{4}-\frac{12\!\cdots\!83}{23\!\cdots\!37}a^{3}+\frac{92\!\cdots\!61}{47\!\cdots\!74}a^{2}+\frac{30\!\cdots\!57}{23\!\cdots\!37}a+\frac{43\!\cdots\!83}{47\!\cdots\!74}$, $\frac{20\!\cdots\!23}{47\!\cdots\!74}a^{14}-\frac{23\!\cdots\!78}{23\!\cdots\!37}a^{13}-\frac{37\!\cdots\!81}{47\!\cdots\!74}a^{12}+\frac{78\!\cdots\!27}{47\!\cdots\!74}a^{11}+\frac{17\!\cdots\!83}{23\!\cdots\!37}a^{10}-\frac{17\!\cdots\!46}{23\!\cdots\!37}a^{9}-\frac{14\!\cdots\!19}{23\!\cdots\!37}a^{8}-\frac{68\!\cdots\!33}{47\!\cdots\!74}a^{7}-\frac{13\!\cdots\!07}{47\!\cdots\!74}a^{6}-\frac{11\!\cdots\!71}{47\!\cdots\!74}a^{5}-\frac{37\!\cdots\!15}{23\!\cdots\!37}a^{4}+\frac{11\!\cdots\!53}{47\!\cdots\!74}a^{3}+\frac{13\!\cdots\!71}{23\!\cdots\!37}a^{2}+\frac{86\!\cdots\!13}{23\!\cdots\!37}a+\frac{36\!\cdots\!59}{23\!\cdots\!37}$, $\frac{15\!\cdots\!84}{23\!\cdots\!37}a^{14}-\frac{47\!\cdots\!05}{23\!\cdots\!37}a^{13}-\frac{22\!\cdots\!50}{23\!\cdots\!37}a^{12}+\frac{73\!\cdots\!96}{23\!\cdots\!37}a^{11}+\frac{18\!\cdots\!32}{23\!\cdots\!37}a^{10}-\frac{72\!\cdots\!41}{47\!\cdots\!74}a^{9}-\frac{34\!\cdots\!87}{47\!\cdots\!74}a^{8}-\frac{76\!\cdots\!05}{47\!\cdots\!74}a^{7}-\frac{13\!\cdots\!77}{47\!\cdots\!74}a^{6}-\frac{27\!\cdots\!53}{23\!\cdots\!37}a^{5}-\frac{71\!\cdots\!67}{47\!\cdots\!74}a^{4}+\frac{15\!\cdots\!99}{47\!\cdots\!74}a^{3}+\frac{32\!\cdots\!55}{47\!\cdots\!74}a^{2}+\frac{47\!\cdots\!13}{47\!\cdots\!74}a-\frac{20\!\cdots\!97}{47\!\cdots\!74}$ (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: | \( 1212792.08283 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 1212792.08283 \cdot 5}{2\cdot\sqrt{7789948783671344586860223}}\cr\approx \mathstrut & 0.340414758803 \end{aligned}\] (assuming GRH)
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: | 15.1.1005792029461681407.1 |
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 | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.3.0.1}{3} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(11\) | 11.5.4.1 | $x^{5} + 55$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
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 | $[\ ]$ |
23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
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 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |