Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-48070x-152415348\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-48070xz^2-152415348z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-62298747x-7110903580074\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{1609967882}{28561}, \frac{64462588287330}{4826809}\right)\)
|
| $\hat{h}(P)$ | ≈ | $17.309330859037391788562031486$ |
Integral points
None
Invariants
| Conductor: | \( 470890 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-10027912931155662760 $ | = | $-1 \cdot 2^{3} \cdot 5 \cdot 7^{10} \cdot 31^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{49}{40} \) | = | $-1 \cdot 2^{-3} \cdot 5^{-1} \cdot 7^{2}$ | 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: | $2.3250311452290097649924102733\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-1.0135542478929911122266325085\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $1.0206100476817066\dots$ | |||
| Szpiro ratio: | $3.920214659623177\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $17.309330859037391788562031486\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.10327251949094055248298608828\dots$ | 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: | $ 6 $ = $ 3\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L'(E,1) $ ≈ $ 10.725469251090466921448002170 $ | 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
$\displaystyle 10.725469251 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.103273 \cdot 17.309331 \cdot 6}{1^2} \approx 10.725469251$
Modular invariants
Modular form 470890.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12700800 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $II^{*}$ | Additive | -1 | 2 | 10 | 0 |
| $31$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 26040 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 16799 & 0 \\ 0 & 26039 \end{array}\right),\left(\begin{array}{rr} 26035 & 6 \\ 26034 & 7 \end{array}\right),\left(\begin{array}{rr} 13021 & 16926 \\ 8463 & 24739 \end{array}\right),\left(\begin{array}{rr} 6511 & 16926 \\ 14973 & 24739 \end{array}\right),\left(\begin{array}{rr} 10417 & 16926 \\ 651 & 24739 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 14879 & 0 \\ 0 & 26039 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1086 & 15841 \\ 5425 & 10851 \end{array}\right)$.
The torsion field $K:=\Q(E[26040])$ is a degree-$3981057196032000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/26040\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 470890.ce
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 490.g2, its twist by $217$.
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.