Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-534753x-150770815\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-534753xz^2-150770815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-43315020x-110041869168\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{5179019}{4225}, \frac{8831271372}{274625}\right)\)
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| $\hat{h}(P)$ | ≈ | $12.630510675241429084552570167$ |
Torsion generators
\( \left(845, 0\right) \)
Integral points
\( \left(845, 0\right) \)
Invariants
| Conductor: | \( 3136 \) | = | $2^{6} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-56593444029595648 $ | = | $-1 \cdot 2^{36} \cdot 7^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{548347731625}{1835008} \) | = | $-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}$ | 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);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $2.0792032355106864195506908480\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.066527390143111802872166294089\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $12.630510675241429084552570167\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.088255985305271234546711476890\dots$ | 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);
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| Tamagawa product: | $ 16 $ = $ 2^{2}\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)
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| Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 4.4588726582087160938048192561 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 4.458872658 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.088256 \cdot 12.630511 \cdot 16}{2^2} \approx 4.458872658$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27648 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{26}^{*}$ | Additive | -1 | 6 | 36 | 18 |
| $7$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 397 & 36 \\ 88 & 29 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 251 & 468 \\ 494 & 143 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 216 & 91 \end{array}\right),\left(\begin{array}{rr} 377 & 468 \\ 242 & 143 \end{array}\right),\left(\begin{array}{rr} 212 & 495 \\ 281 & 326 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$870912$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ss | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 7 | 1,1 | - | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 2 | 0,0 | - | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 3136.z
consists of 6 curves linked by isogenies of
degrees dividing 18.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.7.1-28672.7-e1 |
| $2$ | \(\Q(\sqrt{-42}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | 4.2.1792.1 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.2.6273179136.1 | \(\Z/6\Z\) | Not in database |
| $6$ | 6.0.169375836672.2 | \(\Z/18\Z\) | Not in database |
| $8$ | 8.0.481890304.2 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.157351936.3 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.12745506816.16 | \(\Z/12\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/18\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $12$ | 12.0.28688174048340020035584.5 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | 16.0.162447943996702457856.8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.