Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-277839372x+1744518730736\) | (homogenize, simplify) |
\(y^2z=x^3-277839372xz^2+1744518730736z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-277839372x+1744518730736\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(5365, 639009\right)\) |
$\hat{h}(P)$ | ≈ | $1.2072868661903680611521372952$ |
Torsion generators
\( \left(8452, 0\right) \)
Integral points
\((-10988,\pm 1863000)\), \((5365,\pm 639009)\), \( \left(8452, 0\right) \), \((29032,\pm 4260060)\)
Invariants
Conductor: | \( 463680 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $57929521816327111940505600 $ | = | $2^{21} \cdot 3^{8} \cdot 5^{2} \cdot 7^{12} \cdot 23^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{12411881707829361287041}{303132494474220600} \) | = | $2^{-3} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-12} \cdot 23^{-3} \cdot 29^{3} \cdot 47^{3} \cdot 16987^{3}$ | 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: | $3.7275121854269432153196555682\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.1384852702529704054961847676\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.2072868661903680611521372952\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.062486537909894264675572463897\dots$ | 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: | $ 576 $ = $ 2\cdot2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\cdot3 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 10.863241420654350469664997615 $ | 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
$\displaystyle 10.863241421 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.062487 \cdot 1.207287 \cdot 576}{2^2} \approx 10.863241421$
Modular invariants
Modular form 463680.2.a.ky
For more coefficients, see the Downloads section to the right.
Modular degree: | 191102976 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{11}^{*}$ | Additive | 1 | 6 | 21 | 3 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$7$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$23$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1657 & 12 \\ 1662 & 73 \end{array}\right),\left(\begin{array}{rr} 2749 & 12 \\ 2748 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2710 & 2751 \end{array}\right),\left(\begin{array}{rr} 2750 & 2757 \\ 1407 & 8 \end{array}\right),\left(\begin{array}{rr} 919 & 2748 \\ 2294 & 2687 \end{array}\right),\left(\begin{array}{rr} 1266 & 817 \\ 2645 & 2186 \end{array}\right),\left(\begin{array}{rr} 970 & 3 \\ 693 & 2752 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$98488811520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 463680.ky
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4830.bk1, its twist by $-24$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.