Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760)
gp: K = bnfinit(y^21 - 42*y^19 + 756*y^17 - 7616*y^15 - 20*y^14 + 47040*y^13 + 560*y^12 - 183456*y^11 - 6160*y^10 + 448448*y^9 + 33600*y^8 - 657588*y^7 - 94080*y^6 + 513240*y^5 + 125440*y^4 - 121184*y^3 - 62720*y^2 - 54432*y - 3760, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760)
\( x^{21} - 42 x^{19} + 756 x^{17} - 7616 x^{15} - 20 x^{14} + 47040 x^{13} + 560 x^{12} - 183456 x^{11} + \cdots - 3760 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[7, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-948784467301828652679243568312534649249988608\)
\(\medspace = -\,2^{18}\cdot 7^{30}\cdot 107^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(138.60\) | 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: | $2^{6/7}7^{242/147}107^{1/2}\approx 461.2829834966581$ | ||
| Ramified primes: |
\(2\), \(7\), \(107\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-107}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{2408}a^{14}-\frac{1}{86}a^{12}+\frac{11}{86}a^{10}-\frac{17}{86}a^{8}-\frac{11}{172}a^{7}-\frac{2}{43}a^{6}-\frac{9}{86}a^{5}+\frac{17}{43}a^{4}+\frac{18}{43}a^{3}+\frac{13}{43}a^{2}-\frac{18}{43}a-\frac{8}{301}$, $\frac{1}{2408}a^{15}-\frac{1}{86}a^{13}-\frac{21}{172}a^{11}-\frac{17}{86}a^{9}-\frac{11}{172}a^{8}-\frac{2}{43}a^{7}-\frac{9}{86}a^{6}+\frac{17}{43}a^{5}-\frac{7}{86}a^{4}+\frac{13}{43}a^{3}-\frac{18}{43}a^{2}-\frac{8}{301}a$, $\frac{1}{2408}a^{16}+\frac{9}{172}a^{12}-\frac{5}{43}a^{10}-\frac{11}{172}a^{9}-\frac{7}{86}a^{8}+\frac{9}{86}a^{7}+\frac{4}{43}a^{6}-\frac{1}{86}a^{5}+\frac{16}{43}a^{4}+\frac{13}{43}a^{3}+\frac{132}{301}a^{2}+\frac{12}{43}a+\frac{11}{43}$, $\frac{1}{2408}a^{17}+\frac{9}{172}a^{13}-\frac{5}{43}a^{11}-\frac{11}{172}a^{10}-\frac{7}{86}a^{9}+\frac{9}{86}a^{8}+\frac{4}{43}a^{7}-\frac{1}{86}a^{6}+\frac{16}{43}a^{5}+\frac{13}{43}a^{4}+\frac{132}{301}a^{3}+\frac{12}{43}a^{2}+\frac{11}{43}a$, $\frac{1}{248024}a^{18}-\frac{19}{124012}a^{17}-\frac{9}{62006}a^{16}-\frac{47}{248024}a^{15}+\frac{25}{248024}a^{14}+\frac{693}{8858}a^{13}+\frac{359}{8858}a^{12}-\frac{1403}{17716}a^{11}-\frac{855}{4429}a^{10}-\frac{1265}{8858}a^{9}-\frac{3179}{17716}a^{8}-\frac{3727}{17716}a^{7}-\frac{1455}{8858}a^{6}-\frac{3477}{8858}a^{5}-\frac{7841}{31003}a^{4}+\frac{12715}{31003}a^{3}-\frac{7524}{31003}a^{2}+\frac{11667}{31003}a-\frac{15208}{31003}$, $\frac{1}{248024}a^{19}-\frac{19}{124012}a^{17}+\frac{27}{248024}a^{16}-\frac{5}{124012}a^{15}-\frac{5}{31003}a^{14}+\frac{214}{4429}a^{13}-\frac{75}{17716}a^{12}+\frac{1667}{17716}a^{11}+\frac{1529}{8858}a^{10}-\frac{27}{17716}a^{9}-\frac{670}{4429}a^{8}-\frac{84}{4429}a^{7}-\frac{286}{4429}a^{6}+\frac{9904}{31003}a^{5}-\frac{3733}{8858}a^{4}-\frac{935}{31003}a^{3}+\frac{6945}{31003}a^{2}-\frac{8376}{31003}a+\frac{6312}{31003}$, $\frac{1}{248024}a^{20}+\frac{25}{248024}a^{17}-\frac{39}{248024}a^{16}+\frac{1}{8858}a^{15}-\frac{11}{62006}a^{14}-\frac{143}{17716}a^{13}+\frac{523}{17716}a^{12}+\frac{1553}{17716}a^{11}+\frac{2471}{17716}a^{10}-\frac{1897}{17716}a^{9}-\frac{775}{4429}a^{8}-\frac{1551}{8858}a^{7}+\frac{3128}{31003}a^{6}-\frac{4225}{8858}a^{5}-\frac{3101}{8858}a^{4}-\frac{13143}{31003}a^{3}+\frac{15124}{31003}a^{2}-\frac{94}{4429}a+\frac{1059}{31003}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $13$ | 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: |
$\frac{9}{2408}a^{14}-\frac{9}{86}a^{12}+\frac{99}{86}a^{10}-\frac{270}{43}a^{8}-\frac{13}{172}a^{7}+\frac{756}{43}a^{6}+\frac{91}{86}a^{5}-\frac{1008}{43}a^{4}-\frac{182}{43}a^{3}+\frac{504}{43}a^{2}+\frac{182}{43}a+\frac{1132}{301}$, $\frac{29}{35432}a^{20}-\frac{29}{17716}a^{19}-\frac{261}{8858}a^{18}+\frac{7221}{124012}a^{17}+\frac{54897}{124012}a^{16}-\frac{107155}{124012}a^{15}-\frac{111534}{31003}a^{14}+\frac{61301}{8858}a^{13}+\frac{76072}{4429}a^{12}-\frac{143176}{4429}a^{11}-\frac{218279}{4429}a^{10}+\frac{393817}{4429}a^{9}+\frac{375869}{4429}a^{8}-\frac{615094}{4429}a^{7}-\frac{769879}{8858}a^{6}+\frac{492147}{4429}a^{5}+\frac{208956}{4429}a^{4}-\frac{774810}{31003}a^{3}-\frac{6614}{721}a^{2}-\frac{396076}{31003}a-\frac{23629}{31003}$, $\frac{29}{248024}a^{20}-\frac{319}{124012}a^{19}-\frac{493}{124012}a^{18}+\frac{11803}{124012}a^{17}+\frac{1769}{31003}a^{16}-\frac{182323}{124012}a^{15}-\frac{14239}{31003}a^{14}+\frac{108799}{8858}a^{13}+\frac{10597}{4429}a^{12}-\frac{266275}{4429}a^{11}-\frac{40789}{4429}a^{10}+\frac{774080}{4429}a^{9}+\frac{125185}{4429}a^{8}-\frac{1294768}{4429}a^{7}-\frac{37379}{602}a^{6}+\frac{7895084}{31003}a^{5}+\frac{2313184}{31003}a^{4}-\frac{2116676}{31003}a^{3}-\frac{962274}{31003}a^{2}-\frac{918440}{31003}a-\frac{259363}{31003}$, $\frac{49}{35432}a^{20}+\frac{477}{124012}a^{19}-\frac{6059}{124012}a^{18}-\frac{8523}{62006}a^{17}+\frac{89333}{124012}a^{16}+\frac{255553}{124012}a^{15}-\frac{1419059}{248024}a^{14}-\frac{148855}{8858}a^{13}+\frac{116517}{4429}a^{12}+\frac{1434095}{17716}a^{11}-\frac{307199}{4429}a^{10}-\frac{2079495}{8858}a^{9}+\frac{821377}{8858}a^{8}+\frac{7096321}{17716}a^{7}-\frac{223441}{8858}a^{6}-\frac{22823777}{62006}a^{5}-\frac{4738039}{62006}a^{4}+\frac{3856064}{31003}a^{3}+\frac{1969893}{31003}a^{2}+\frac{756013}{31003}a+\frac{123884}{31003}$, $\frac{1055}{248024}a^{20}-\frac{2313}{248024}a^{19}-\frac{1409}{8858}a^{18}+\frac{85749}{248024}a^{17}+\frac{620499}{248024}a^{16}-\frac{1336453}{248024}a^{15}-\frac{5329339}{248024}a^{14}+\frac{811055}{17716}a^{13}+\frac{485394}{4429}a^{12}-\frac{2035685}{8858}a^{11}-\frac{6006963}{17716}a^{10}+\frac{12228219}{17716}a^{9}+\frac{11133587}{17716}a^{8}-\frac{21185765}{17716}a^{7}-\frac{41841467}{62006}a^{6}+\frac{33155489}{31003}a^{5}+\frac{1721821}{4429}a^{4}-\frac{9528812}{31003}a^{3}-\frac{3063947}{31003}a^{2}-\frac{3117549}{31003}a-\frac{211896}{31003}$, $\frac{799}{35432}a^{20}-\frac{101}{2408}a^{19}-\frac{222743}{248024}a^{18}+\frac{52035}{31003}a^{17}+\frac{3729237}{248024}a^{16}-\frac{7033413}{248024}a^{15}-\frac{33916775}{248024}a^{14}+\frac{4637261}{17716}a^{13}+\frac{3219241}{4429}a^{12}-\frac{6300772}{4429}a^{11}-\frac{9969027}{4429}a^{10}+\frac{80546943}{17716}a^{9}+\frac{67228125}{17716}a^{8}-\frac{140905761}{17716}a^{7}-\frac{27041337}{8858}a^{6}+\frac{192473625}{31003}a^{5}+\frac{84377313}{62006}a^{4}-\frac{27411402}{31003}a^{3}-\frac{20708598}{31003}a^{2}-\frac{21131535}{31003}a-\frac{1472096}{31003}$, $\frac{253}{124012}a^{20}-\frac{235}{35432}a^{19}-\frac{2587}{35432}a^{18}+\frac{29803}{124012}a^{17}+\frac{68329}{62006}a^{16}-\frac{905721}{248024}a^{15}-\frac{1141431}{124012}a^{14}+\frac{535713}{17716}a^{13}+\frac{209259}{4429}a^{12}-\frac{2627937}{17716}a^{11}-\frac{1402485}{8858}a^{10}+\frac{1940939}{4429}a^{9}+\frac{6423739}{17716}a^{8}-\frac{6685279}{8858}a^{7}-\frac{17703282}{31003}a^{6}+\frac{5957909}{8858}a^{5}+\frac{2435191}{4429}a^{4}-\frac{4969063}{31003}a^{3}-\frac{6865790}{31003}a^{2}-\frac{3990457}{31003}a-\frac{1490621}{31003}$, $\frac{661}{248024}a^{20}+\frac{239}{62006}a^{19}-\frac{589}{5768}a^{18}-\frac{5243}{35432}a^{17}+\frac{410401}{248024}a^{16}+\frac{85223}{35432}a^{15}-\frac{84927}{5768}a^{14}-\frac{381787}{17716}a^{13}+\frac{1381323}{17716}a^{12}+\frac{2054423}{17716}a^{11}-\frac{4383749}{17716}a^{10}-\frac{6763881}{17716}a^{9}+\frac{7840747}{17716}a^{8}+\frac{13191247}{17716}a^{7}-\frac{22135455}{62006}a^{6}-\frac{23826534}{31003}a^{5}-\frac{532314}{31003}a^{4}+\frac{1211571}{4429}a^{3}+\frac{4673909}{31003}a^{2}+\frac{3660}{43}a+\frac{192622}{31003}$, $\frac{2167}{248024}a^{20}-\frac{617}{248024}a^{19}-\frac{44991}{124012}a^{18}+\frac{3073}{35432}a^{17}+\frac{198169}{31003}a^{16}-\frac{75583}{62006}a^{15}-\frac{7694429}{124012}a^{14}+\frac{151617}{17716}a^{13}+\frac{148321}{412}a^{12}-\frac{497939}{17716}a^{11}-\frac{22384115}{17716}a^{10}+\frac{81501}{8858}a^{9}+\frac{11388170}{4429}a^{8}+\frac{1790083}{8858}a^{7}-\frac{83052673}{31003}a^{6}-\frac{14774437}{31003}a^{5}+\frac{58195537}{62006}a^{4}+\frac{1115216}{4429}a^{3}+\frac{9599479}{31003}a^{2}-\frac{2710}{301}a+\frac{35163}{31003}$, $\frac{25}{35432}a^{20}-\frac{1971}{124012}a^{19}-\frac{4205}{124012}a^{18}+\frac{19357}{35432}a^{17}+\frac{82751}{124012}a^{16}-\frac{69165}{8858}a^{15}-\frac{1776147}{248024}a^{14}+\frac{1064481}{17716}a^{13}+\frac{828731}{17716}a^{12}-\frac{2380739}{8858}a^{11}-\frac{3467015}{17716}a^{10}+\frac{3099605}{4429}a^{9}+\frac{2354334}{4429}a^{8}-\frac{17478141}{17716}a^{7}-\frac{4015898}{4429}a^{6}+\frac{17093091}{31003}a^{5}+\frac{26681875}{31003}a^{4}+\frac{1039872}{4429}a^{3}-\frac{9907528}{31003}a^{2}-\frac{1483615}{4429}a-\frac{1365474}{31003}$, $\frac{227}{248024}a^{20}-\frac{1135}{62006}a^{19}-\frac{3015}{124012}a^{18}+\frac{160547}{248024}a^{17}+\frac{30103}{124012}a^{16}-\frac{592867}{62006}a^{15}-\frac{269289}{248024}a^{14}+\frac{338762}{4429}a^{13}+\frac{37501}{17716}a^{12}-\frac{6365199}{17716}a^{11}-\frac{111477}{17716}a^{10}+\frac{8907227}{8858}a^{9}+\frac{243093}{4429}a^{8}-\frac{28804879}{17716}a^{7}-\frac{13084609}{62006}a^{6}+\frac{42536400}{31003}a^{5}+\frac{20387205}{62006}a^{4}-\frac{254858}{721}a^{3}-\frac{5469008}{31003}a^{2}-\frac{4671855}{31003}a-\frac{339788}{31003}$, $\frac{12317}{248024}a^{20}-\frac{19945}{248024}a^{19}-\frac{500765}{248024}a^{18}+\frac{809391}{248024}a^{17}+\frac{8641869}{248024}a^{16}-\frac{6965883}{124012}a^{15}-\frac{41149937}{124012}a^{14}+\frac{2356351}{4429}a^{13}+\frac{8405189}{4429}a^{12}-\frac{13306018}{4429}a^{11}-\frac{117806699}{17716}a^{10}+\frac{181004313}{17716}a^{9}+\frac{62052770}{4429}a^{8}-\frac{88720043}{4429}a^{7}-\frac{1049045999}{62006}a^{6}+\frac{1243626843}{62006}a^{5}+\frac{660661671}{62006}a^{4}-\frac{210543496}{31003}a^{3}-\frac{52728905}{31003}a^{2}-\frac{93048289}{31003}a-\frac{6724803}{31003}$, $\frac{151}{248024}a^{20}-\frac{41}{248024}a^{19}-\frac{2671}{124012}a^{18}+\frac{1093}{248024}a^{17}+\frac{78299}{248024}a^{16}-\frac{9719}{248024}a^{15}-\frac{614517}{248024}a^{14}+\frac{1031}{17716}a^{13}+\frac{49805}{4429}a^{12}+\frac{11251}{8858}a^{11}-\frac{528553}{17716}a^{10}-\frac{170071}{17716}a^{9}+\frac{790735}{17716}a^{8}+\frac{543075}{17716}a^{7}-\frac{1940121}{62006}a^{6}-\frac{1473697}{31003}a^{5}-\frac{219120}{31003}a^{4}+\frac{850527}{31003}a^{3}+\frac{572771}{31003}a^{2}+\frac{148557}{31003}a+\frac{8226}{31003}$
| 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: | \( 903623889948000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 903623889948000 \cdot 1}{2\cdot\sqrt{948784467301828652679243568312534649249988608}}\cr\approx \mathstrut & 0.725843469280345 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 20*x^14 + 47040*x^13 + 560*x^12 - 183456*x^11 - 6160*x^10 + 448448*x^9 + 33600*x^8 - 657588*x^7 - 94080*x^6 + 513240*x^5 + 125440*x^4 - 121184*x^3 - 62720*x^2 - 54432*x - 3760);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_7^2:(C_3\times S_3)$ (as 21T26):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 882 |
| The 20 conjugacy class representatives for $C_7^2:(C_3\times S_3)$ |
| Character table for $C_7^2:(C_3\times S_3)$ |
Intermediate fields
| 3.1.107.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 14 sibling: | data not computed |
| Degree 21 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 14.0.52481615129370406177120250117500928.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $21$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $21$ | ${\href{/padicField/13.3.0.1}{3} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21$ | ${\href{/padicField/41.3.0.1}{3} }^{7}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }^{7}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.14.12.1 | $x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 397 x^{7} + 371 x^{6} + 224 x^{5} + 21 x^{4} + 7 x^{3} + 70 x^{2} + 35 x + 7$ | $7$ | $2$ | $12$ | $(C_7:C_3) \times C_2$ | $[\ ]_{7}^{6}$ | |
|
\(7\)
| 7.7.10.3 | $x^{7} + 14 x^{4} + 7$ | $7$ | $1$ | $10$ | $C_7:C_3$ | $[5/3]_{3}$ |
| 7.14.20.19 | $x^{14} - 196 x^{12} - 154 x^{11} - 980 x^{10} + 980 x^{9} + 1225 x^{8} + 14 x^{7} - 1372 x^{5} - 1078 x^{4} + 49$ | $7$ | $2$ | $20$ | 14T14 | $[5/3, 5/3]_{3}^{2}$ | |
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.3.0.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 107.3.0.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 107.6.3.2 | $x^{6} + 331 x^{4} + 210 x^{3} + 34372 x^{2} - 66360 x + 1124253$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 107.6.3.2 | $x^{6} + 331 x^{4} + 210 x^{3} + 34372 x^{2} - 66360 x + 1124253$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |