Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-1356x-16222\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-1356xz^2-16222z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1756755x-751571730\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 414050 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $47302728200 $ | = | $2^{3} \cdot 5^{2} \cdot 7^{2} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{46585}{8} \) | = | $2^{-3} \cdot 5 \cdot 7 \cdot 11^{3}$ | 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: | $0.76869318349642716220454837390\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-1.1063395054825768191065473597\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.8304580690764805\dots$ | |||
Szpiro ratio: | $2.57075068454062\dots$ |
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: | $0.79603736750163037679236969547\dots$ | 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: | $ 2 $ = $ 1\cdot1\cdot1\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: | $1$ | 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) $ ≈ $ 1.5920747350032607535847393909 $ | 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
$\displaystyle 1.592074735 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.796037 \cdot 1.000000 \cdot 2}{1^2} \approx 1.592074735$
Modular invariants
Modular form 414050.2.a.l
For more coefficients, see the Downloads section to the right.
Modular degree: | 497664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$13$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 |
---|---|---|
$2$ | 2G | 8.2.0.2 |
$3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 32743 & 18 \\ 32742 & 19 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 26729 & 22698 \\ 29016 & 22985 \end{array}\right),\left(\begin{array}{rr} 20801 & 6318 \\ 8658 & 29485 \end{array}\right),\left(\begin{array}{rr} 21971 & 22698 \\ 6318 & 24935 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 32696 & 32701 \end{array}\right),\left(\begin{array}{rr} 27730 & 27729 \\ 21411 & 5032 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 17639 & 0 \\ 0 & 32759 \end{array}\right),\left(\begin{array}{rr} 8191 & 22698 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$1051769626951680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 414050l
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2450x1, its twist by $13$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.