Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4479852x+3269382064\)
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(homogenize, simplify) |
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\(y^2z=x^3-4479852xz^2+3269382064z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4479852x+3269382064\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1560, 8788)$ | $2.8575984053968241338888637306$ | $\infty$ |
| $(884, 0)$ | $0$ | $2$ |
Integral points
\((-2275,\pm 41067)\), \( \left(884, 0\right) \), \((1560,\pm 8788)\)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $1136427663619420323840$ | = | $2^{19} \cdot 3^{12} \cdot 5 \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{10779215329}{1232010} \) | = | $2^{-1} \cdot 3^{-6} \cdot 5^{-1} \cdot 13^{-2} \cdot 47^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7724577572407876290685398186$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.099043836663953548781674702832$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0867566045656807$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.39533806345372$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8575984053968241338888637306$ |
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| Real period: | $\Omega$ | ≈ | $0.14949595666980507937315748622$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4175952591432616800241258728 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.417595259 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.149496 \cdot 2.857598 \cdot 32}{2^2} \\ & \approx 3.417595259\end{aligned}$$
Modular invariants
Modular form 486720.2.a.kg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24772608 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 521 & 4 \\ 1042 & 9 \end{array}\right),\left(\begin{array}{rr} 314 & 1 \\ 623 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 4 \\ 602 & 9 \end{array}\right),\left(\begin{array}{rr} 1557 & 4 \\ 1556 & 5 \end{array}\right),\left(\begin{array}{rr} 977 & 586 \\ 584 & 975 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 486720kg
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 390e2, its twist by $-312$.
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.