Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-309196917x-4918825913009\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-309196917xz^2-4918825913009z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-4947150675x-314809805583250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{91571}{4}, -\frac{91571}{8}\right) \)
Integral points
None
Invariants
| Conductor: | \( 450450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-8560658494301226784252031250 $ | = | $-1 \cdot 2 \cdot 3^{30} \cdot 5^{7} \cdot 7 \cdot 11^{3} \cdot 13^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{286999819333751016766729}{751553009101890965970} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-24} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-3} \cdot 13^{-4} \cdot 37^{3} \cdot 47^{3} \cdot 83^{3} \cdot 457^{3}$ | 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: | $4.0473473331614888207736164820\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $2.6933222326103837877756141969\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $0.9919358401725701\dots$ | |||
| Szpiro ratio: | $5.5286023869325955\dots$ | |||
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: | $0.016735763399672270108252508097\dots$ | 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: | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\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: | $2$ | 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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L(E,1) $ ≈ $ 0.53554442878951264346408025911 $ | 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
$\displaystyle 0.535544429 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.016736 \cdot 1.000000 \cdot 32}{2^2} \approx 0.535544429$
Modular invariants
Modular form 450450.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 382205952 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{24}^{*}$ | Additive | -1 | 2 | 30 | 24 |
| $5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120120 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 24008 & 120099 \\ 285 & 374 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 118814 & 110891 \end{array}\right),\left(\begin{array}{rr} 36961 & 24 \\ 83172 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 120097 & 24 \\ 120096 & 25 \end{array}\right),\left(\begin{array}{rr} 21856 & 3 \\ 108741 & 120034 \end{array}\right),\left(\begin{array}{rr} 115122 & 95119 \\ 25517 & 6692 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 60059 & 120096 \\ 80080 & 100099 \end{array}\right),\left(\begin{array}{rr} 30046 & 21 \\ 29745 & 119746 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 102976 & 3 \\ 33861 & 120034 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$64274810535936000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 450450e
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30030r4, its twist by $-15$.
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 $0$.