Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-265953324x-477847035344\)
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(homogenize, simplify) |
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\(y^2z=x^3-265953324xz^2-477847035344z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-265953324x-477847035344\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-27668776456/4165681, 8483454052495092/8502154921)$ | $23.239935217212762880202198308$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $1105274229777894709011677184$ | = | $2^{30} \cdot 3^{11} \cdot 7^{2} \cdot 17^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{222165413800219579417}{118033833938006016} \) | = | $2^{-12} \cdot 3^{-5} \cdot 7 \cdot 13^{3} \cdot 17^{-9} \cdot 23^{3} \cdot 10589^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8805727252654667716154300621$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9672274519156084109410671375$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0717635926093514$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.336697735268672$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $23.239935217212762880202198308$ |
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| Real period: | $\Omega$ | ≈ | $0.039710726024233094771876863545$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.6914988009266482521987037077 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.691498801 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.039711 \cdot 23.239935 \cdot 4}{1^2} \\ & \approx 3.691498801\end{aligned}$$
Modular invariants
Modular form 479808.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 278691840 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{20}^{*}$ | additive | 1 | 6 | 30 | 12 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $17$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
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 \( 2856 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 122 & 1311 \\ 1 & 358 \end{array}\right),\left(\begin{array}{rr} 2141 & 2850 \\ 2139 & 2837 \end{array}\right),\left(\begin{array}{rr} 1595 & 2850 \\ 1929 & 2837 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1433 & 2850 \\ 1434 & 2849 \end{array}\right),\left(\begin{array}{rr} 2851 & 6 \\ 2850 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1227 & 2 \\ 1234 & 7 \end{array}\right),\left(\begin{array}{rr} 1427 & 0 \\ 0 & 2855 \end{array}\right)$.
The torsion field $K:=\Q(E[2856])$ is a degree-$727720132608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2856\Z)$.
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 | $2$ | \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
| $7$ | additive | $14$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 479808.o
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4998.a1, its twist by $-24$.
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.