Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1366779x-586716410\)
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(homogenize, simplify) |
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\(y^2z=x^3-1366779xz^2-586716410z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1366779x-586716410\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12944426360792046/40122093025, 1472723980790024610729094/8036655843372625)$ | $32.732067606992380775349214761$ | $\infty$ |
| $(-790, 0)$ | $0$ | $2$ |
| $(-553, 0)$ | $0$ | $2$ |
Integral points
\( \left(-790, 0\right) \), \( \left(-553, 0\right) \), \( \left(1343, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 449352 \) | = | $2^{3} \cdot 3^{2} \cdot 79^{2}$ |
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| Discriminant: | $\Delta$ | = | $14698568868924957696$ | = | $2^{10} \cdot 3^{10} \cdot 79^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1556068}{81} \) | = | $2^{2} \cdot 3^{-4} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4352513747676809748337007731$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.87640134626650570913142138398$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0321229303519712$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.148688014190353$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $32.732067606992380775349214761$ |
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| Real period: | $\Omega$ | ≈ | $0.14008074431517593194645255767$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.1702647867243044278206593772 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.170264787 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.140081 \cdot 32.732068 \cdot 32}{4^2} \\ & \approx 9.170264787\end{aligned}$$
Modular invariants
Modular form 449352.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7907328 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $79$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.48.0.88 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1896 = 2^{3} \cdot 3 \cdot 79 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 1106 \\ 316 & 159 \end{array}\right),\left(\begin{array}{rr} 1107 & 1738 \\ 1106 & 475 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1892 & 1893 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 1895 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1889 & 8 \\ 1888 & 9 \end{array}\right),\left(\begin{array}{rr} 631 & 1264 \\ 316 & 1263 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1896])$ is a degree-$14765137920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1896\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 56169 = 3^{2} \cdot 79^{2} \) |
| $3$ | additive | $8$ | \( 49928 = 2^{3} \cdot 79^{2} \) |
| $79$ | additive | $3122$ | \( 72 = 2^{3} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 449352i
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 24a2, its twist by $237$.
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.