Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-452119x-115761838\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-452119xz^2-115761838z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-585946251x-5392195111674\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{1693}{4}, \frac{1689}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 5547 \) | = | $3 \cdot 43^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $144455554505315187 $ | = | $3^{12} \cdot 43^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{1616855892553}{22851963} \) | = | $3^{-12} \cdot 11^{6} \cdot 43^{-1} \cdot 97^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $2.0974269719326573264487207165\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.21682691408587611471229945983\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0580583409717772\dots$ | |||
Szpiro ratio: | $5.878509647047474\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.18427142239799878490846816907\dots$ | 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: | $ 4 $ = $ 2\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)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L(E,1) $ ≈ $ 0.18427142239799878490846816907 $ | 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
$\displaystyle 0.184271422 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.184271 \cdot 1.000000 \cdot 4}{2^2} \approx 0.184271422$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 55440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$43$ | $2$ | $I_{1}^{*}$ | 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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1032 = 2^{3} \cdot 3 \cdot 43 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 904 & 395 \\ 907 & 936 \end{array}\right),\left(\begin{array}{rr} 1025 & 8 \\ 1024 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1026 & 1027 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 664 & 1029 \\ 931 & 1030 \end{array}\right),\left(\begin{array}{rr} 689 & 8 \\ 692 & 33 \end{array}\right),\left(\begin{array}{rr} 907 & 906 \\ 658 & 139 \end{array}\right)$.
The torsion field $K:=\Q(E[1032])$ is a degree-$5126381568$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1032\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 5547a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 129b3, its twist by $-43$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-43}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{43}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{43})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | 4.2.20353792.1 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.414276848779264.6 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.70892257536.3 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.33556424751120384.22 | \(\Z/8\Z\) | Not in database |
$8$ | 8.2.13824820988163.2 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 43 |
---|---|---|---|
Reduction type | ord | nonsplit | add |
$\lambda$-invariant(s) | 5 | 0 | - |
$\mu$-invariant(s) | 2 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 $0$.