Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-44309772x-122714547696\)
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(homogenize, simplify) |
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\(y^2z=x^3-44309772xz^2-122714547696z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-44309772x-122714547696\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-937693082298926487306240$ | = | $-1 \cdot 2^{53} \cdot 3^{6} \cdot 5 \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{1762712152495281}{171798691840} \) | = | $-1 \cdot 2^{-35} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{2} \cdot 7283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3408124668581326116785629925$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89680243253031422317059637800$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1377369369986685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.932203719583798$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.029097026325597250870030190660$ |
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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)$ | ≈ | $5.7030171598170611705259173694 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $49$ = $7^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.703017160 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{49 \cdot 0.029097 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 5.703017160\end{aligned}$$
Modular invariants
Modular form 486720.2.a.ps
For more coefficients, see the Downloads section to the right.
| Modular degree: | 72253440 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{43}^{*}$ | additive | -1 | 6 | 53 | 35 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 913 & 528 \\ 1806 & 10753 \end{array}\right),\left(\begin{array}{rr} 9811 & 3642 \\ 9030 & 10831 \end{array}\right),\left(\begin{array}{rr} 10907 & 14 \\ 10906 & 15 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 736 & 7287 \\ 8001 & 3634 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 3632 & 3633 \\ 10017 & 7286 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 3632 & 3633 \\ 1827 & 7286 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $62$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 486720ps
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 1690g2, its twist by $24$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.