Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-6072038x-5393754844\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-6072038xz^2-5393754844z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-7869360627x-251627417908146\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-1380, 19612\right)\) |
$\hat{h}(P)$ | ≈ | $0.93596845926911731554462367057$ |
Torsion generators
\( \left(-1119, 559\right) \), \( \left(2825, -1413\right) \)
Integral points
\( \left(-1425, 19837\right) \), \( \left(-1425, -18413\right) \), \( \left(-1380, 19612\right) \), \( \left(-1380, -18233\right) \), \( \left(-1119, 559\right) \), \( \left(2825, -1413\right) \), \( \left(2850, 19837\right) \), \( \left(2850, -22688\right) \), \( \left(6450, 469837\right) \), \( \left(6450, -476288\right) \), \( \left(7755, 639487\right) \), \( \left(7755, -647243\right) \), \( \left(717675, 607621087\right) \), \( \left(717675, -608338763\right) \)
Invariants
Conductor: | \( 428910 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $1762283982538139062500 $ | = | $2^{2} \cdot 3^{8} \cdot 5^{8} \cdot 17^{2} \cdot 29^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{41623544884956481}{2962701562500} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-8} \cdot 17^{-2} \cdot 346561^{3}$ | 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: | $2.8238142016264010037746749646\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.1401662866331639901830389484\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0054946838642063\dots$ | |||
Szpiro ratio: | $4.508536248656377\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.93596845926911731554462367057\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.096607404714106479056429095291\dots$ | 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: | $ 1024 $ = $ 2\cdot2^{3}\cdot2^{3}\cdot2\cdot2^{2} $ | 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: | $4$ | 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) $ ≈ $ 5.7869749596320193396963246073 $ | 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
$\displaystyle 5.786974960 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.096607 \cdot 0.935968 \cdot 1024}{4^2} \approx 5.786974960$
Modular invariants
Modular form 428910.2.a.bh
For more coefficients, see the Downloads section to the right.
Modular degree: | 25690112 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$29$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 8.48.0.91 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 118320 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \cdot 29 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 53039 & 0 \\ 0 & 118319 \end{array}\right),\left(\begin{array}{rr} 118305 & 16 \\ 118304 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 69369 & 20416 \\ 114434 & 113565 \end{array}\right),\left(\begin{array}{rr} 1 & 69368 \\ 0 & 73951 \end{array}\right),\left(\begin{array}{rr} 93033 & 69368 \\ 8932 & 8121 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 78881 & 69368 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3133 & 69368 \\ 52490 & 32597 \end{array}\right)$.
The torsion field $K:=\Q(E[118320])$ is a degree-$39393916511846400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/118320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 428910.bh
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 510.e3, its twist by $29$.
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.