Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-430823x-67411618\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-430823xz^2-67411618z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-558346635x-3136781241594\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-253, 5173\right)\) |
$\hat{h}(P)$ | ≈ | $5.4504602168988954074002908057$ |
Torsion generators
\( \left(-\frac{669}{4}, \frac{665}{8}\right) \)
Integral points
\( \left(-253, 5173\right) \), \( \left(-253, -4921\right) \)
Invariants
Conductor: | \( 17787 \) | = | $3 \cdot 7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $3165014301887765943 $ | = | $3^{16} \cdot 7^{3} \cdot 11^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{14553591673375}{5208653241} \) | = | $3^{-16} \cdot 5^{3} \cdot 11^{-2} \cdot 19^{3} \cdot 257^{3}$ | 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: | $2.2507533340848189776572247275\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.56532816042180537934991475265\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.4504602168988954074002908057\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.19187817960536931947186915293\dots$ | 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: | $ 8 $ = $ 2\cdot2\cdot2 $ | 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) $ ≈ $ 2.0916487688600929403920231483 $ | 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
$\displaystyle 2.091648769 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.191878 \cdot 5.450460 \cdot 8}{2^2} \approx 2.091648769$
Modular invariants
Modular form 17787.2.a.k
For more coefficients, see the Downloads section to the right.
Modular degree: | 307200 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
$7$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$11$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
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 | 8.12.0.27 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 616 = 2^{3} \cdot 7 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 608 & 595 \end{array}\right),\left(\begin{array}{rr} 609 & 8 \\ 608 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 162 \\ 154 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 360 & 3 \\ 221 & 600 \end{array}\right),\left(\begin{array}{rr} 466 & 465 \\ 21 & 478 \end{array}\right),\left(\begin{array}{rr} 285 & 2 \\ 22 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[616])$ is a degree-$851558400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/616\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 17787i
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1617d2, its twist by $-11$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.166012.1 | \(\Z/4\Z\) | Not in database |
$8$ | 8.4.112885695053824.25 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.440959746304.7 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.7055355940864.44 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.440959746304.9 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | nonsplit | ss | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | 1 | 3,1 | - | - | 1 | 1 | 1 | 1 | 1 | 3 | 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.