Properties

Label 18.2.476...777.1
Degree $18$
Signature $[2, 8]$
Discriminant $4.769\times 10^{27}$
Root discriminant \(34.49\)
Ramified primes $7,17$
Class number $1$
Class group trivial
Galois group $D_{18}$ (as 18T13)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)
 
gp: K = bnfinit(y^18 - 2*y^17 + 17*y^15 - 17*y^14 + 51*y^13 + 119*y^12 - 476*y^11 + 493*y^10 + 1632*y^9 - 3638*y^8 - 3502*y^7 + 6936*y^6 + 1972*y^5 - 3451*y^4 + 3264*y^3 + 17*y^2 - 750*y + 225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)
 

\( x^{18} - 2 x^{17} + 17 x^{15} - 17 x^{14} + 51 x^{13} + 119 x^{12} - 476 x^{11} + 493 x^{10} + 1632 x^{9} - 3638 x^{8} - 3502 x^{7} + 6936 x^{6} + 1972 x^{5} - 3451 x^{4} + \cdots + 225 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4768875488962616064464333777\) \(\medspace = 7^{8}\cdot 17^{17}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.49\)
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:  $7^{1/2}17^{17/18}\approx 38.427282044851594$
Ramified primes:   \(7\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{9}+\frac{1}{15}a^{8}-\frac{2}{15}a^{7}+\frac{7}{15}a^{6}+\frac{1}{15}a^{5}-\frac{1}{3}a^{4}+\frac{7}{15}a^{3}+\frac{4}{15}a^{2}+\frac{1}{5}a$, $\frac{1}{15}a^{10}+\frac{2}{15}a^{8}-\frac{1}{15}a^{7}-\frac{1}{15}a^{6}-\frac{1}{15}a^{5}+\frac{2}{15}a^{4}+\frac{2}{15}a^{3}+\frac{4}{15}a^{2}+\frac{2}{15}a$, $\frac{1}{15}a^{11}+\frac{2}{15}a^{8}-\frac{7}{15}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{2}{15}a^{4}-\frac{1}{3}a^{3}-\frac{1}{15}a^{2}-\frac{1}{15}a$, $\frac{1}{195}a^{12}+\frac{4}{195}a^{11}+\frac{1}{39}a^{10}+\frac{4}{195}a^{9}-\frac{4}{65}a^{8}+\frac{11}{65}a^{7}-\frac{17}{65}a^{6}+\frac{18}{65}a^{5}+\frac{38}{195}a^{4}+\frac{8}{195}a^{3}-\frac{62}{195}a^{2}-\frac{73}{195}a-\frac{3}{13}$, $\frac{1}{195}a^{13}+\frac{2}{195}a^{11}-\frac{1}{65}a^{10}-\frac{2}{195}a^{9}+\frac{29}{195}a^{8}-\frac{79}{195}a^{7}-\frac{28}{195}a^{6}-\frac{7}{39}a^{5}+\frac{38}{195}a^{4}-\frac{27}{65}a^{3}-\frac{7}{195}a^{2}+\frac{1}{15}a-\frac{1}{13}$, $\frac{1}{975}a^{14}-\frac{1}{975}a^{12}-\frac{1}{65}a^{11}-\frac{2}{65}a^{10}-\frac{3}{325}a^{9}-\frac{32}{195}a^{8}+\frac{68}{975}a^{7}+\frac{79}{975}a^{6}-\frac{397}{975}a^{5}-\frac{4}{25}a^{4}+\frac{476}{975}a^{3}-\frac{217}{975}a^{2}+\frac{7}{195}a-\frac{6}{13}$, $\frac{1}{4875}a^{15}+\frac{2}{4875}a^{14}-\frac{1}{4875}a^{13}+\frac{1}{1625}a^{12}+\frac{4}{975}a^{11}+\frac{32}{1625}a^{10}+\frac{97}{4875}a^{9}-\frac{167}{4875}a^{8}-\frac{102}{325}a^{7}+\frac{41}{4875}a^{6}+\frac{34}{75}a^{5}-\frac{896}{4875}a^{4}+\frac{283}{975}a^{3}+\frac{376}{4875}a^{2}-\frac{28}{65}a-\frac{11}{65}$, $\frac{1}{102375}a^{16}-\frac{1}{14625}a^{15}+\frac{2}{34125}a^{14}-\frac{188}{102375}a^{13}-\frac{1}{14625}a^{12}+\frac{397}{34125}a^{11}-\frac{467}{102375}a^{10}+\frac{562}{20475}a^{9}+\frac{103}{1125}a^{8}+\frac{199}{2625}a^{7}+\frac{341}{102375}a^{6}+\frac{21514}{102375}a^{5}+\frac{1249}{4875}a^{4}-\frac{25009}{102375}a^{3}+\frac{18841}{102375}a^{2}-\frac{179}{1365}a+\frac{6}{35}$, $\frac{1}{313358232607875}a^{17}+\frac{1404856129}{313358232607875}a^{16}+\frac{25409243719}{313358232607875}a^{15}+\frac{155108268703}{313358232607875}a^{14}+\frac{7330690499}{4820895886275}a^{13}-\frac{209454488186}{313358232607875}a^{12}+\frac{41219692024}{3443497061625}a^{11}-\frac{2913813250412}{313358232607875}a^{10}-\frac{739245317434}{24104479431375}a^{9}-\frac{1195848720691}{313358232607875}a^{8}+\frac{61169396576297}{313358232607875}a^{7}-\frac{3361738081379}{8953092360225}a^{6}+\frac{186254065181}{777563852625}a^{5}+\frac{4692836456396}{12534329304315}a^{4}+\frac{11845255351616}{44765461801125}a^{3}+\frac{6362826393968}{44765461801125}a^{2}+\frac{39793089398}{134777734455}a+\frac{351822543028}{1392703256035}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Trivial group, which has order $1$

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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{267491154221}{313358232607875}a^{17}-\frac{554320101008}{313358232607875}a^{16}+\frac{3024157772}{8034826477125}a^{15}+\frac{4410102035996}{313358232607875}a^{14}-\frac{4945784171834}{313358232607875}a^{13}+\frac{5221212374426}{104452744202625}a^{12}+\frac{28878812742842}{313358232607875}a^{11}-\frac{124888144421648}{313358232607875}a^{10}+\frac{154619313560258}{313358232607875}a^{9}+\frac{127173194556781}{104452744202625}a^{8}-\frac{189554544285202}{62671646521575}a^{7}-\frac{728864082056752}{313358232607875}a^{6}+\frac{3339024985532}{673888672275}a^{5}+\frac{298299557428762}{313358232607875}a^{4}-\frac{96387412811423}{62671646521575}a^{3}+\frac{166702789556143}{104452744202625}a^{2}+\frac{869702132}{10367518035}a+\frac{105352959437}{1392703256035}$, $\frac{26852169814}{104452744202625}a^{17}-\frac{748335121}{1606965295425}a^{16}+\frac{89054677}{278540651207}a^{15}+\frac{16642535171}{4178109768105}a^{14}-\frac{418639133749}{104452744202625}a^{13}+\frac{637637760023}{34817581400875}a^{12}+\frac{702411857246}{20890548840525}a^{11}-\frac{9779770502759}{104452744202625}a^{10}+\frac{16619469548726}{104452744202625}a^{9}+\frac{11674219472038}{34817581400875}a^{8}-\frac{86821478330378}{104452744202625}a^{7}-\frac{44792413425889}{104452744202625}a^{6}+\frac{4212733718693}{3369443361375}a^{5}-\frac{186239610105071}{104452744202625}a^{4}-\frac{47853906475451}{34817581400875}a^{3}+\frac{29870585912208}{6963516280175}a^{2}+\frac{377435310754}{134777734455}a-\frac{235949549816}{278540651207}$, $\frac{1866495444}{2678275492375}a^{17}-\frac{3525882872}{1790618472045}a^{16}+\frac{542908405316}{313358232607875}a^{15}+\frac{983651001859}{104452744202625}a^{14}-\frac{58769033324}{3443497061625}a^{13}+\frac{15238950907631}{313358232607875}a^{12}+\frac{209746370668}{6963516280175}a^{11}-\frac{100905850593602}{313358232607875}a^{10}+\frac{24436000630532}{44765461801125}a^{9}+\frac{199130168637121}{313358232607875}a^{8}-\frac{90526719702429}{34817581400875}a^{7}-\frac{327450673007722}{313358232607875}a^{6}+\frac{7429546108663}{1444047154875}a^{5}-\frac{3938191553056}{104452744202625}a^{4}-\frac{980451923850331}{313358232607875}a^{3}+\frac{431440007053681}{313358232607875}a^{2}-\frac{1540681117}{10367518035}a-\frac{19766344656}{198957608005}$, $\frac{6514546781}{2984364120075}a^{17}-\frac{1126296696743}{313358232607875}a^{16}-\frac{73310786482}{44765461801125}a^{15}+\frac{1307923556048}{34817581400875}a^{14}-\frac{7572254365271}{313358232607875}a^{13}+\frac{333735797726}{3443497061625}a^{12}+\frac{31753914513379}{104452744202625}a^{11}-\frac{297537006932699}{313358232607875}a^{10}+\frac{45288040582741}{62671646521575}a^{9}+\frac{13869045314626}{3443497061625}a^{8}-\frac{702547048953271}{104452744202625}a^{7}-\frac{32\!\cdots\!33}{313358232607875}a^{6}+\frac{132963479145013}{10108330084125}a^{5}+\frac{154912548914183}{14921820600375}a^{4}-\frac{23\!\cdots\!58}{313358232607875}a^{3}+\frac{833075483657707}{313358232607875}a^{2}+\frac{14369499002}{3455839345}a-\frac{1436948785784}{1392703256035}$, $\frac{22300096991}{313358232607875}a^{17}-\frac{4103331968}{6963516280175}a^{16}+\frac{45324708523}{313358232607875}a^{15}+\frac{64260550988}{44765461801125}a^{14}-\frac{820747973018}{104452744202625}a^{13}+\frac{4546157537}{62671646521575}a^{12}-\frac{1354891300244}{62671646521575}a^{11}-\frac{1033442734417}{8034826477125}a^{10}+\frac{2012243428718}{20890548840525}a^{9}+\frac{279373679219}{12534329304315}a^{8}-\frac{317709990577037}{313358232607875}a^{7}+\frac{3691771922719}{104452744202625}a^{6}+\frac{25075891523057}{10108330084125}a^{5}+\frac{512052411364348}{313358232607875}a^{4}+\frac{50683877719487}{34817581400875}a^{3}+\frac{181603293265988}{313358232607875}a^{2}-\frac{195260071264}{134777734455}a+\frac{663259592139}{1392703256035}$, $\frac{198283711414}{44765461801125}a^{17}-\frac{210911305973}{24104479431375}a^{16}+\frac{46614206689}{44765461801125}a^{15}+\frac{22523873661376}{313358232607875}a^{14}-\frac{22277236822361}{313358232607875}a^{13}+\frac{10949302075843}{44765461801125}a^{12}+\frac{154317872711356}{313358232607875}a^{11}-\frac{24989153950283}{12534329304315}a^{10}+\frac{690004447393007}{313358232607875}a^{9}+\frac{295597386960308}{44765461801125}a^{8}-\frac{184601280432067}{12534329304315}a^{7}-\frac{46\!\cdots\!47}{313358232607875}a^{6}+\frac{50441064043459}{2021666016825}a^{5}+\frac{446925952820636}{44765461801125}a^{4}-\frac{546668165400368}{62671646521575}a^{3}+\frac{38\!\cdots\!06}{313358232607875}a^{2}+\frac{3656734240}{26955546891}a-\frac{1336014646732}{1392703256035}$, $\frac{102611790773}{44765461801125}a^{17}-\frac{1247721140242}{313358232607875}a^{16}-\frac{91490586712}{44765461801125}a^{15}+\frac{12699506910422}{313358232607875}a^{14}-\frac{10021285421752}{313358232607875}a^{13}+\frac{4467451244768}{44765461801125}a^{12}+\frac{96227579762948}{313358232607875}a^{11}-\frac{5326238868821}{4820895886275}a^{10}+\frac{249255361292383}{313358232607875}a^{9}+\frac{183998804962573}{44765461801125}a^{8}-\frac{24\!\cdots\!44}{313358232607875}a^{7}-\frac{33\!\cdots\!56}{313358232607875}a^{6}+\frac{140061372032509}{10108330084125}a^{5}+\frac{434246666277283}{44765461801125}a^{4}-\frac{15\!\cdots\!24}{313358232607875}a^{3}+\frac{67748038027324}{12534329304315}a^{2}+\frac{54787259903}{26955546891}a-\frac{443067397299}{278540651207}$, $\frac{29865326492}{10108330084125}a^{17}-\frac{50487910931}{10108330084125}a^{16}-\frac{86919057}{44925911485}a^{15}+\frac{101940060766}{2021666016825}a^{14}-\frac{353403781727}{10108330084125}a^{13}+\frac{151091910442}{1123147787125}a^{12}+\frac{4028475801524}{10108330084125}a^{11}-\frac{13213858696307}{10108330084125}a^{10}+\frac{10348092343874}{10108330084125}a^{9}+\frac{5938044268022}{1123147787125}a^{8}-\frac{18762389384314}{2021666016825}a^{7}-\frac{5526023087077}{404333203365}a^{6}+\frac{2325541038199}{134777734455}a^{5}+\frac{25466579290382}{2021666016825}a^{4}-\frac{15069981369218}{2021666016825}a^{3}+\frac{17143023523844}{3369443361375}a^{2}+\frac{17074976701}{10367518035}a-\frac{73342427014}{44925911485}$, $\frac{34321461198}{34817581400875}a^{17}-\frac{27853879909}{24104479431375}a^{16}-\frac{60855917056}{62671646521575}a^{15}+\frac{327834164102}{20890548840525}a^{14}-\frac{1015314300817}{313358232607875}a^{13}+\frac{14716490583094}{313358232607875}a^{12}+\frac{758858950391}{4973940200125}a^{11}-\frac{106521876312907}{313358232607875}a^{10}+\frac{2268943877611}{12534329304315}a^{9}+\frac{545068490935229}{313358232607875}a^{8}-\frac{71719524278144}{34817581400875}a^{7}-\frac{238785809927894}{44765461801125}a^{6}+\frac{22400248237166}{10108330084125}a^{5}+\frac{460437465330616}{104452744202625}a^{4}+\frac{38985447426997}{44765461801125}a^{3}+\frac{146646828663527}{44765461801125}a^{2}+\frac{302754219412}{134777734455}a+\frac{1198067645357}{1392703256035}$ Copy content Toggle raw display
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:  \( 10997947.525 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 10997947.525 \cdot 1}{2\cdot\sqrt{4768875488962616064464333777}}\cr\approx \mathstrut & 0.77369999480 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)
 
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^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225, 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^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225);
 
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^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225);
 
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

$D_{18}$ (as 18T13):

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 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.1.2023.1, 6.2.69572993.1, 9.1.16748793615841.1

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

Galois closure: data not computed
Degree 18 sibling: 18.0.33382128422738312451250336439.1
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 ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{9}$ ${\href{/padicField/5.2.0.1}{2} }^{9}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ $18$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(17\) Copy content Toggle raw display 17.18.17.1$x^{18} + 17$$18$$1$$17$$D_{18}$$[\ ]_{18}^{2}$

Additional information

This field is associated with the 17-torsion points on the elliptic curves from the isogeny class 49.a1, a curve with complex multiplication by $\Q(\sqrt{-7})$.