Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-2306582082x+42432281377812\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-2306582082xz^2+42432281377812z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36905313315x+2715629102866654\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(104547/4, -104547/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 489762 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $7598226556215870947240214528$ | = | $2^{15} \cdot 3^{8} \cdot 7^{12} \cdot 13^{6} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{385693937170561837203625}{2159357734550274048} \) | = | $2^{-15} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{-12} \cdot 23^{-2} \cdot 59^{3} \cdot 443^{3} \cdot 557^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1921427220181782112879108710$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3603618989533549975635445318$ |
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| $abc$ quality: | $Q$ | ≈ | $1.041884065687328$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.822973163368752$ | |||
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: | $\Omega$ | ≈ | $0.041928309747318468286804786999$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6834118238283819703555063679 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.683411824 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.041928 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.683411824\end{aligned}$$
Modular invariants
Modular form 489762.2.a.bm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 530841600 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{15}$ | nonsplit multiplicative | 1 | 1 | 15 | 15 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 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 | 8.6.0.6 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7176 = 2^{3} \cdot 3 \cdot 13 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 7165 & 12 \\ 7164 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 7126 & 7167 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 7175 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5617 & 6084 \\ 4446 & 625 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2770 & 3315 \\ 2457 & 4408 \end{array}\right),\left(\begin{array}{rr} 4783 & 1092 \\ 1196 & 7175 \end{array}\right),\left(\begin{array}{rr} 1275 & 3016 \\ 4550 & 4109 \end{array}\right)$.
The torsion field $K:=\Q(E[7176])$ is a degree-$5377489108992$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7176\Z)$.
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 | $4$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 3887 = 13^{2} \cdot 23 \) |
| $5$ | good | $2$ | \( 244881 = 3^{2} \cdot 7 \cdot 13^{2} \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 69966 = 2 \cdot 3^{2} \cdot 13^{2} \cdot 23 \) |
| $13$ | additive | $86$ | \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 21294 = 2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 489762.bm
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 966.f1, its twist by $-39$.
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 $0$.