Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-157441x+23998303\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-157441xz^2+23998303z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-12752748x+17533021104\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 27584 \) | = | $2^{6} \cdot 431$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-118472377892864 $ | = | $-1 \cdot 2^{38} \cdot 431 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1646417855125441}{451936256} \) | = | $-1 \cdot 2^{-20} \cdot 431^{-1} \cdot 118081^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.6829504622871631520170890021\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.64322969144724518789124081991\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9455399461306241\dots$ | |||
Szpiro ratio: | $4.646899052305992\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.57630857255770381118973433687\dots$ | 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: | $ 4 $ = $ 2^{2}\cdot1 $ | 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: | $1$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.3052342902308152447589373475 $ | 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
$\displaystyle 2.305234290 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.576309 \cdot 1.000000 \cdot 4}{1^2} \approx 2.305234290$
Modular invariants
Modular form 27584.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 122880 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{28}^{*}$ | Additive | 1 | 6 | 38 | 20 |
$431$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17240 = 2^{3} \cdot 5 \cdot 431 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 6 & 13 \\ 17185 & 17121 \end{array}\right),\left(\begin{array}{rr} 4311 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15474 & 17235 \\ 12965 & 12934 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 17239 & 17230 \\ 0 & 14653 \end{array}\right),\left(\begin{array}{rr} 8619 & 0 \\ 0 & 17239 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 4255 & 4191 \end{array}\right),\left(\begin{array}{rr} 4301 & 10 \\ 4300 & 11 \end{array}\right),\left(\begin{array}{rr} 17231 & 10 \\ 17230 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[17240])$ is a degree-$528797196288000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17240\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 27584.s
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 862.d1, its twist by $8$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/5\Z\) | Not in database |
$3$ | 3.1.431.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.80062991.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.95109632.2 | \(\Z/10\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | Not in database |
$20$ | 20.0.46460754422900124442403525470303643851649220608000000000000000.1 | \(\Z/5\Z\) | 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 | 431 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | - | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
$\mu$-invariant(s) | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.