Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-342965101x-2444714009152\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-342965101xz^2-2444714009152z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-444482770275x-114059243362673250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-42777/4, 42773/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 17850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $280328660493706125000$ | = | $2^{3} \cdot 3^{8} \cdot 5^{6} \cdot 7^{2} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{285531136548675601769470657}{17941034271597192} \) | = | $2^{-3} \cdot 3^{-8} \cdot 7^{-2} \cdot 17^{-8} \cdot 658492993^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3838479255400679498351867878$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5791289693230177625348071212$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0624727966937526$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.208862069364124$ | |||
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: | $\Omega$ | ≈ | $0.035082220509399623105410419620$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 1\cdot2^{3}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.2452621126015758787462668557 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.245262113 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.035082 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 2.245262113\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3932160 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $2$ | $I_{8}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.48.0.217 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1360 = 2^{4} \cdot 5 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 1262 & 1347 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 1359 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1345 & 16 \\ 1344 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1356 & 1357 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 76 & 345 \\ 955 & 1226 \end{array}\right),\left(\begin{array}{rr} 1271 & 560 \\ 1070 & 841 \end{array}\right),\left(\begin{array}{rr} 241 & 560 \\ 840 & 401 \end{array}\right)$.
The torsion field $K:=\Q(E[1360])$ is a degree-$4812963840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1360\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 2975 = 5^{2} \cdot 7 \cdot 17 \) |
| $5$ | additive | $14$ | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 17850.t
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714.f1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.4.6294077440000.28 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6294077440000.75 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.13684080640000.43 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | split | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 5 | 1 | - | 0 | 0 |
| $\mu$-invariant(s) | 3 | 0 | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.