Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-150789x+22487149\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-150789xz^2+22487149z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-12213936x+16429773402\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{5596}{25}, \frac{69}{125}\right)\) |
$\hat{h}(P)$ | ≈ | $4.1920939722134483582688785163$ |
Integral points
None
Invariants
Conductor: | \( 59584 \) | = | $2^{6} \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-143061184 $ | = | $-1 \cdot 2^{6} \cdot 7^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{50357871050752}{19} \) | = | $-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$ | 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: | $1.3529678217989471107813873160\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.033439156991317803520094883549\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.104947099482495\dots$ | |||
Szpiro ratio: | $4.3095867678427\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $4.1920939722134483582688785163\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.1030118380976675437335846361\dots$ | 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);
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Tamagawa product: | $ 1 $ | 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)
|
Torsion order: | $1$ | 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) $ ≈ $ 4.6239292777693081233386916944 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 4.623929278 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.103012 \cdot 4.192094 \cdot 1}{1^2} \approx 4.623929278$
Modular invariants
Modular form 59584.2.a.u
For more coefficients, see the Downloads section to the right.
Modular degree: | 163296 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II$ | Additive | 1 | 6 | 6 | 0 |
$7$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 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 |
---|---|---|
$3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28728 = 2^{3} \cdot 3^{3} \cdot 7 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 11339 & 24570 \\ 11340 & 24569 \end{array}\right),\left(\begin{array}{rr} 23687 & 24549 \\ 7581 & 28636 \end{array}\right),\left(\begin{array}{rr} 28675 & 54 \\ 28674 & 55 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 12242 & 20475 \\ 28343 & 3632 \end{array}\right),\left(\begin{array}{rr} 14363 & 0 \\ 0 & 28727 \end{array}\right),\left(\begin{array}{rr} 24623 & 0 \\ 0 & 28727 \end{array}\right),\left(\begin{array}{rr} 4073 & 16380 \\ 24290 & 28307 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 22906 & 21967 \end{array}\right),\left(\begin{array}{rr} 7181 & 0 \\ 0 & 28727 \end{array}\right)$.
The torsion field $K:=\Q(E[28728])$ is a degree-$92643856220160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28728\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 59584bn
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a2, its twist by $-56$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{42}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.76.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.205978074624.4 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.450474049202688.2 | \(\Z/9\Z\) | Not in database |
$6$ | 6.2.6846916608.2 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | 12.0.1557130999647530092068864.1 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.201906434092298129641985063519986384896.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.2112012614563679138031447702469448035763264421888.1 | \(\Z/18\Z\) | 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 | add | ord | ord | add | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | 7 | 1 | - | 1 | 1 | 1 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | - | 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.