Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-48020008x+128063959988\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-48020008xz^2+128063959988z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3889620675x+93370295693250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(17228, 2101050)$ | $6.7742257770575451022929518281$ | $\infty$ |
| $(3998, 0)$ | $0$ | $2$ |
| $(4003, 0)$ | $0$ | $2$ |
Integral points
\( \left(-8002, 0\right) \), \( \left(3998, 0\right) \), \( \left(4003, 0\right) \), \((17228,\pm 2101050)\)
Invariants
| Conductor: | $N$ | = | \( 8400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $8301313440000000000$ | = | $2^{14} \cdot 3^{2} \cdot 5^{10} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{191342053882402567201}{129708022500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{-8} \cdot 73^{3} \cdot 193^{3} \cdot 409^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9457219633925858889620339747$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4478558266155903922444221866$ |
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| $abc$ quality: | $Q$ | ≈ | $1.05504045380176$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.15749016772393$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.7742257770575451022929518281$ |
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| Real period: | $\Omega$ | ≈ | $0.19270562375312912046562798937$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2217256152496001483955850836 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.221725615 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.192706 \cdot 6.774226 \cdot 64}{4^2} \\ & \approx 5.221725615\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
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 |
|---|---|---|
| $2$ | 2Cs | 16.96.0.72 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 489 & 8 \\ 1156 & 1641 \end{array}\right),\left(\begin{array}{rr} 327 & 1664 \\ 604 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 6 & 517 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1679 & 1664 \\ 1674 & 113 \end{array}\right),\left(\begin{array}{rr} 9 & 16 \\ 754 & 345 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 8400.ce
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210.e2, its twist by $-20$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.3317760000.9 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7644119040000.46 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.7644119040000.25 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.12960000.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.11007531417600000000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.16.633846573131471257600000000.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | 16.0.259916710394723932569600000000.4 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | add | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.