Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-14700x-236882000\) | (homogenize, simplify) |
\(y^2z=x^3-14700xz^2-236882000z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-14700x-236882000\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(202566, 91169536\right)\) |
$\hat{h}(P)$ | ≈ | $8.9282580086449906301118356497$ |
Integral points
\((202566,\pm 91169536)\)
Invariants
Conductor: | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-24240648093696000000 $ | = | $-1 \cdot 2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 29 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{117649}{8118144} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7^{6} \cdot 29^{-1}$ | 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: | $2.3984372852504025035773025163\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.0046914138593795064534520490400\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.2436350520132442\dots$ | |||
Szpiro ratio: | $4.024738582534854\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $8.9282580086449906301118356497\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.097251720796802655842458383908\dots$ | 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: | $ 8 $ = $ 2^{2}\cdot2\cdot1\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)
|
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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 6.9463076404684792119782968711 $ | 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
$\displaystyle 6.946307640 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.097252 \cdot 8.928258 \cdot 8}{1^2} \approx 6.946307640$
Modular invariants
Modular form 417600.2.a.fx
For more coefficients, see the Downloads section to the right.
Modular degree: | 6021120 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{15}^{*}$ | Additive | 1 | 6 | 25 | 7 |
$3$ | $2$ | $I_{7}^{*}$ | Additive | -1 | 2 | 13 | 7 |
$5$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$29$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24360 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 29 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 18271 & 14630 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12776 & 19495 \\ 10745 & 4866 \end{array}\right),\left(\begin{array}{rr} 6091 & 2450 \\ 0 & 13051 \end{array}\right),\left(\begin{array}{rr} 9743 & 0 \\ 0 & 24359 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21104 & 4865 \\ 11375 & 19494 \end{array}\right),\left(\begin{array}{rr} 24347 & 14 \\ 24346 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 4864 & 4865 \\ 7315 & 19494 \end{array}\right)$.
The torsion field $K:=\Q(E[24360])$ is a degree-$506907013939200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 417600.fx
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 174.e2, its twist by $-120$.
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.