Normalized defining polynomial
\( x^{15} - x^{14} - 2 x^{13} + x^{12} + x^{11} - x^{10} + 2 x^{9} - 15 x^{8} + 8 x^{7} + 28 x^{6} + \cdots + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-154972454814106259\) \(\medspace = -\,11^{13}\cdot 67^{2}\) | 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: | \(14.00\) | 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: | $11^{9/10}67^{2/3}\approx 142.76988787487005$ | ||
Ramified primes: | \(11\), \(67\) | 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: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1375967}a^{14}+\frac{175735}{1375967}a^{13}-\frac{613357}{1375967}a^{12}+\frac{221128}{1375967}a^{11}+\frac{90195}{1375967}a^{10}-\frac{631321}{1375967}a^{9}-\frac{232077}{1375967}a^{8}-\frac{621807}{1375967}a^{7}-\frac{79672}{1375967}a^{6}+\frac{601628}{1375967}a^{5}-\frac{230139}{1375967}a^{4}+\frac{90720}{1375967}a^{3}-\frac{559682}{1375967}a^{2}+\frac{597135}{1375967}a-\frac{6898}{1375967}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{9871118}{1375967}a^{14}-\frac{5442576}{1375967}a^{13}-\frac{22352934}{1375967}a^{12}+\frac{195017}{1375967}a^{11}+\frac{10168561}{1375967}a^{10}-\frac{5609891}{1375967}a^{9}+\frac{16901521}{1375967}a^{8}-\frac{140513656}{1375967}a^{7}+\frac{15248762}{1375967}a^{6}+\frac{286186857}{1375967}a^{5}-\frac{211351518}{1375967}a^{4}-\frac{166885954}{1375967}a^{3}+\frac{198831449}{1375967}a^{2}+\frac{18484429}{1375967}a-\frac{24636408}{1375967}$, $\frac{2954835}{1375967}a^{14}-\frac{1377570}{1375967}a^{13}-\frac{6917210}{1375967}a^{12}-\frac{439608}{1375967}a^{11}+\frac{3046529}{1375967}a^{10}-\frac{1390323}{1375967}a^{9}+\frac{5191165}{1375967}a^{8}-\frac{41374953}{1375967}a^{7}+\frac{707811}{1375967}a^{6}+\frac{87320377}{1375967}a^{5}-\frac{57031774}{1375967}a^{4}-\frac{55546486}{1375967}a^{3}+\frac{57685609}{1375967}a^{2}+\frac{8146219}{1375967}a-\frac{7132494}{1375967}$, $\frac{1450324}{1375967}a^{14}-\frac{431204}{1375967}a^{13}-\frac{3712168}{1375967}a^{12}-\frac{390954}{1375967}a^{11}+\frac{1542424}{1375967}a^{10}-\frac{645425}{1375967}a^{9}+\frac{2204592}{1375967}a^{8}-\frac{19723503}{1375967}a^{7}-\frac{3384903}{1375967}a^{6}+\frac{46596970}{1375967}a^{5}-\frac{25687417}{1375967}a^{4}-\frac{33722369}{1375967}a^{3}+\frac{30118715}{1375967}a^{2}+\frac{5591940}{1375967}a-\frac{3806796}{1375967}$, $a$, $\frac{5494597}{1375967}a^{14}-\frac{2846191}{1375967}a^{13}-\frac{12822567}{1375967}a^{12}+\frac{113142}{1375967}a^{11}+\frac{5893959}{1375967}a^{10}-\frac{3138561}{1375967}a^{9}+\frac{9205248}{1375967}a^{8}-\frac{77583185}{1375967}a^{7}+\frac{5248701}{1375967}a^{6}+\frac{164217103}{1375967}a^{5}-\frac{117463343}{1375967}a^{4}-\frac{100794874}{1375967}a^{3}+\frac{115621493}{1375967}a^{2}+\frac{10484392}{1375967}a-\frac{15854728}{1375967}$, $\frac{13996050}{1375967}a^{14}-\frac{8460232}{1375967}a^{13}-\frac{31332111}{1375967}a^{12}+\frac{1378211}{1375967}a^{11}+\frac{14821072}{1375967}a^{10}-\frac{8036830}{1375967}a^{9}+\frac{24834469}{1375967}a^{8}-\frac{200082968}{1375967}a^{7}+\frac{33016177}{1375967}a^{6}+\frac{404291653}{1375967}a^{5}-\frac{312370017}{1375967}a^{4}-\frac{224334716}{1375967}a^{3}+\frac{287797026}{1375967}a^{2}+\frac{20588044}{1375967}a-\frac{35803487}{1375967}$, $\frac{5483797}{1375967}a^{14}-\frac{3325698}{1375967}a^{13}-\frac{12472105}{1375967}a^{12}+\frac{609454}{1375967}a^{11}+\frac{5972595}{1375967}a^{10}-\frac{2788246}{1375967}a^{9}+\frac{10000941}{1375967}a^{8}-\frac{78162512}{1375967}a^{7}+\frac{12606761}{1375967}a^{6}+\frac{161198943}{1375967}a^{5}-\frac{122462413}{1375967}a^{4}-\frac{88498667}{1375967}a^{3}+\frac{114188095}{1375967}a^{2}+\frac{7831787}{1375967}a-\frac{12906612}{1375967}$, $\frac{10978394}{1375967}a^{14}-\frac{6171889}{1375967}a^{13}-\frac{25294672}{1375967}a^{12}+\frac{722596}{1375967}a^{11}+\frac{11866554}{1375967}a^{10}-\frac{5926807}{1375967}a^{9}+\frac{19206189}{1375967}a^{8}-\frac{155745697}{1375967}a^{7}+\frac{17855462}{1375967}a^{6}+\frac{325416046}{1375967}a^{5}-\frac{239925756}{1375967}a^{4}-\frac{189293541}{1375967}a^{3}+\frac{229809588}{1375967}a^{2}+\frac{18316179}{1375967}a-\frac{28761340}{1375967}$, $\frac{17550747}{1375967}a^{14}-\frac{10049003}{1375967}a^{13}-\frac{39734123}{1375967}a^{12}+\frac{1252205}{1375967}a^{11}+\frac{18022284}{1375967}a^{10}-\frac{10259379}{1375967}a^{9}+\frac{30689452}{1375967}a^{8}-\frac{249594030}{1375967}a^{7}+\frac{32980535}{1375967}a^{6}+\frac{511898243}{1375967}a^{5}-\frac{386785103}{1375967}a^{4}-\frac{288151214}{1375967}a^{3}+\frac{363226421}{1375967}a^{2}+\frac{23216490}{1375967}a-\frac{43251288}{1375967}$ | 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: | \( 492.79424719 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 492.79424719 \cdot 1}{2\cdot\sqrt{154972454814106259}}\cr\approx \mathstrut & 0.19613617318 \end{aligned}\]
Galois group
$C_7^3:C_6$ (as 15T44):
A solvable group of order 2430 |
The 39 conjugacy class representatives for $C_7^3:C_6$ |
Character table for $C_7^3:C_6$ |
Intermediate fields
\(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 45 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $15$ | $15$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
\(67\) | 67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.3.2.1 | $x^{3} + 67$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |