Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-10875981078x-436571405239668\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-10875981078xz^2-436571405239668z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-14095271477763x-20368464053789787138\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{8243155377459577448466637684705411130049390553029143391522486604101973839189200669177075677661510806665506434723494738329}{67779665206853747420752620106350821796510249501084892202334518415058695811966510138765205777500547653195009157357824}, \frac{3474972445126811645515999011006740684523282546462424174632324948759784613202184862660557680763697946119626814792108665256429574366738002122505069647793414320476983482316697266012309}{558019183117110812100850606751119663427151108560402571131710775707401606375724757941659829142425888767413640648630415827656353156599696307039101626081157513775881705337720832}\right)\) |
$\hat{h}(P)$ | ≈ | $278.01706057723995353078604678$ |
Integral points
None
Invariants
Conductor: | \( 424830 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-6973082191981377389400 $ | = | $-1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{8} \cdot 17^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{12242088317612041}{600} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-2} \cdot 7 \cdot 17^{2} \cdot 18223^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $4.1150544465835169997242874572\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.45676989383312806277944011334\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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||
$abc$ quality: | $1.014544776657247\dots$ | |||
Szpiro ratio: | $6.245875101350812\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $278.01706057723995353078604678\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.0073918381341951943363642454680\dots$ | 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\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);
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Special value: | $ L'(E,1) $ ≈ $ 4.1101142206633953932291296547 $ | 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
$\displaystyle 4.110114221 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.007392 \cdot 278.017061 \cdot 2}{1^2} \approx 4.110114221$
Modular invariants
Modular form 424830.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 349780032 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $1$ | $IV^{*}$ | Additive | 1 | 2 | 8 | 0 |
$17$ | $1$ | $II^{*}$ | Additive | 1 | 2 | 10 | 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 \( 408 = 2^{3} \cdot 3 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 290 & 51 \\ 289 & 154 \end{array}\right),\left(\begin{array}{rr} 167 & 0 \\ 0 & 407 \end{array}\right),\left(\begin{array}{rr} 403 & 6 \\ 402 & 7 \end{array}\right),\left(\begin{array}{rr} 103 & 102 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 205 & 102 \\ 255 & 307 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 237 & 238 \\ 374 & 305 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[408])$ is a degree-$360972288$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/408\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 424830s
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 424830g2, its twist by $-119$.
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.