Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3343371736x+74392891609936\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3343371736xz^2+74392891609936z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-270813110643x+54231605544311442\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1673562/49, 70920998/343)$ | $12.419287450772802697992861771$ | $\infty$ |
| $(32969, 0)$ | $0$ | $2$ |
Integral points
\( \left(32969, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 485520 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $1103215917972669483080448000$ | = | $2^{11} \cdot 3^{7} \cdot 5^{3} \cdot 7^{10} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{83609231549925663172082}{22317062975463375} \) | = | $2 \cdot 3^{-7} \cdot 5^{-3} \cdot 7^{-10} \cdot 17^{-2} \cdot 34706281^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1728653277310252757357501031$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1208737401896340353118533495$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0097143368729002$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.911898208508616$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.419287450772802697992861771$ |
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| Real period: | $\Omega$ | ≈ | $0.047829393705653069232661287529$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.3760279561067553650970518035 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.376027956 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047829 \cdot 12.419287 \cdot 16}{2^2} \\ & \approx 2.376027956\end{aligned}$$
Modular invariants
Modular form 485520.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 371589120 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $17$ | $4$ | $I_{2}^{*}$ | 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 \( 14280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 14277 & 4 \\ 14276 & 5 \end{array}\right),\left(\begin{array}{rr} 5356 & 8929 \\ 1785 & 12496 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 5714 & 1 \\ 11423 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8401 & 4 \\ 2522 & 9 \end{array}\right),\left(\begin{array}{rr} 4762 & 1 \\ 4759 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 7139 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6121 & 4 \\ 12242 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$465740884869120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14280\Z)$.
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 | $4$ | \( 4335 = 3 \cdot 5 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 32368 = 2^{4} \cdot 7 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 13872 = 2^{4} \cdot 3 \cdot 17^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 23120 = 2^{4} \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 485520t
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 14280j2, its twist by $-68$.
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.