Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-165171020x+679236591917\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-165171020xz^2+679236591917z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-214061641947x+31693673357117046\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{42475}{9}, \frac{1819687}{27}\right)\) |
$\hat{h}(P)$ | ≈ | $6.0391850327257445169246785229$ |
Torsion generators
\( \left(-\frac{58221}{4}, \frac{58217}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 60690 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $89041957977184105581092160 $ | = | $2^{6} \cdot 3^{24} \cdot 5 \cdot 7^{4} \cdot 17^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{20645800966247918737249}{3688936444974392640} \) | = | $2^{-6} \cdot 3^{-24} \cdot 5^{-1} \cdot 7^{-4} \cdot 17^{-1} \cdot 97^{3} \cdot 277^{3} \cdot 1021^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.6985177020715840603607563567\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $2.2819110300434760202359890478\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0111769184766262\dots$ | |||
Szpiro ratio: | $6.2088224918747645\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $6.0391850327257445169246785229\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.057525034382802520677135524957\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: | $ 96 $ = $ ( 2 \cdot 3 )\cdot2\cdot1\cdot2\cdot2^{2} $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 8.3377038396397156649867977168 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 8.337703840 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.057525 \cdot 6.039185 \cdot 96}{2^2} \approx 8.337703840$
Modular invariants
Modular form 60690.2.a.bo
For more coefficients, see the Downloads section to the right.
Modular degree: | 21233664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$3$ | $2$ | $I_{24}$ | Non-split multiplicative | 1 | 1 | 24 | 24 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$17$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2B | 4.12.0.8 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 6121 & 24 \\ 2052 & 289 \end{array}\right),\left(\begin{array}{rr} 10711 & 24 \\ 10710 & 1 \end{array}\right),\left(\begin{array}{rr} 2872 & 21 \\ 13995 & 13906 \end{array}\right),\left(\begin{array}{rr} 8384 & 14277 \\ 459 & 86 \end{array}\right),\left(\begin{array}{rr} 14257 & 24 \\ 14256 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6561 & 4168 \\ 2516 & 509 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 12974 & 5051 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 7141 & 24 \\ 4760 & 2381 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$14554402652160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14280\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 60690.bo
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 3570.w5, its twist by $17$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{85}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-85}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{17}) \) | \(\Z/6\Z\) | Not in database |
$4$ | 4.2.39304000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{85})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{17})\) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-5}, \sqrt{17})\) | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.57528168586875.2 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.24716870656000000.26 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.12834170944000000.16 | \(\Z/8\Z\) | Not in database |
$8$ | 8.4.1544804416000000.6 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.13363360000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.6.18809529066767607156978854235133205620800000000.1 | \(\Z/18\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | nonsplit | split | nonsplit | ss | ord | add | ord | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 10 | 1 | 2 | 1 | 1,1 | 1 | - | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0,0 | 0 | - | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.