Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-81912x+8676864\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-81912xz^2+8676864z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1310595x+554008702\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(136, 128)$ | $3.0688079444701822843612094679$ | $\infty$ |
| $(192, -96)$ | $0$ | $2$ |
Integral points
\( \left(-315, 1932\right) \), \( \left(-315, -1617\right) \), \( \left(136, 128\right) \), \( \left(136, -264\right) \), \( \left(192, -96\right) \)
Invariants
| Conductor: | $N$ | = | \( 404586 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $2803242925933632$ | = | $2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 13^{6} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{466385893875}{21509824} \) | = | $2^{-6} \cdot 3^{3} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 19^{-3} \cdot 47^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7256301916674555104018331717$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.16850244076965971952627814169$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9359465023128135$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.528398608687412$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0688079444701822843612094679$ |
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| Real period: | $\Omega$ | ≈ | $0.44812144956384413857414914745$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5007946580360759778454819520 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.500794658 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.448121 \cdot 3.068808 \cdot 16}{2^2} \\ & \approx 5.500794658\end{aligned}$$
Modular invariants
Modular form 404586.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2654208 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 41496 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 3706 & 22347 \\ 23829 & 15952 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 41446 & 41487 \end{array}\right),\left(\begin{array}{rr} 13443 & 3328 \\ 35750 & 38429 \end{array}\right),\left(\begin{array}{rr} 29641 & 6396 \\ 15054 & 38377 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 20749 & 6396 \\ 3198 & 38377 \end{array}\right),\left(\begin{array}{rr} 35111 & 0 \\ 0 & 41495 \end{array}\right),\left(\begin{array}{rr} 41485 & 12 \\ 41484 & 13 \end{array}\right),\left(\begin{array}{rr} 5864 & 12779 \\ 29757 & 3160 \end{array}\right)$.
The torsion field $K:=\Q(E[41496])$ is a degree-$4995905728020480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/41496\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 9633 = 3 \cdot 13^{2} \cdot 19 \) |
| $3$ | additive | $6$ | \( 1183 = 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 57798 = 2 \cdot 3^{2} \cdot 13^{2} \cdot 19 \) |
| $13$ | additive | $86$ | \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 21294 = 2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 404586.ba
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2394.d3, its twist by $-39$.
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.