Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-30651094712x+2065451256913968\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-30651094712xz^2+2065451256913968z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2482738671699x+1505721414506297742\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(101079, 0)$ | $0$ | $2$ |
Integral points
\( \left(101079, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 482664 \) | = | $2^{3} \cdot 3 \cdot 7 \cdot 13^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $42020443894560768$ | = | $2^{11} \cdot 3^{6} \cdot 7^{3} \cdot 13^{6} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{322159999717985454060440834}{4250799} \) | = | $2 \cdot 3^{-6} \cdot 7^{-3} \cdot 11^{6} \cdot 13^{3} \cdot 17^{-1} \cdot 191^{3} \cdot 1811^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1734819842378616474954371355$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2556223899938100791695639700$ |
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| $abc$ quality: | $Q$ | ≈ | $1.108049552899778$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.4224740065651496$ | |||
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: | $\Omega$ | ≈ | $0.083124977740689842359831735092$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 1\cdot( 2 \cdot 3 )\cdot3\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.9849983973296686499078849267 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.984998397 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.083125 \cdot 1.000000 \cdot 72}{2^2} \\ & \approx 5.984998397\end{aligned}$$
Modular invariants
Modular form 482664.2.a.cq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 318504960 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | 1 | 3 | 11 | 0 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12376 = 2^{3} \cdot 7 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8984 & 11427 \\ 4901 & 3810 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 6280 & 11427 \\ 117 & 3810 \end{array}\right),\left(\begin{array}{rr} 12369 & 8 \\ 12368 & 9 \end{array}\right),\left(\begin{array}{rr} 6072 & 10361 \\ 8203 & 4382 \end{array}\right),\left(\begin{array}{rr} 7736 & 7267 \\ 10595 & 5864 \end{array}\right),\left(\begin{array}{rr} 3807 & 0 \\ 0 & 12375 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 12370 & 12371 \end{array}\right)$.
The torsion field $K:=\Q(E[12376])$ is a degree-$132445064134656$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12376\Z)$.
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 | $4$ | \( 20111 = 7 \cdot 13^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 22984 = 2^{3} \cdot 13^{2} \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 68952 = 2^{3} \cdot 3 \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $86$ | \( 2856 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28392 = 2^{3} \cdot 3 \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 482664cq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2856h4, its twist by $13$.
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 $0$.