Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-2888008x+1770038512\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-2888008xz^2+1770038512z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-233928675x+1289656289250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(772, 0)$ | $0$ | $2$ |
| $(1177, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1948, 0\right) \), \( \left(772, 0\right) \), \( \left(1177, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 20400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $189612900000000000000$ | = | $2^{14} \cdot 3^{8} \cdot 5^{14} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{41623544884956481}{2962701562500} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-8} \cdot 17^{-2} \cdot 346561^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6380324234101594869006507365$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1401662866331639901830389484$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0054946838642063$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.667660121489651$ | |||
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: | $\Omega$ | ≈ | $0.17579225040765274110450605656$ |
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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)$ | ≈ | $0.70316900163061096441802422623 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.703169002 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.175792 \cdot 1.000000 \cdot 64}{4^2} \\ & \approx 0.703169002\end{aligned}$$
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 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
| $17$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.48.0.91 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 4065 & 16 \\ 4064 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 807 & 4072 \\ 3308 & 39 \end{array}\right),\left(\begin{array}{rr} 4079 & 4072 \\ 0 & 3569 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1361 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3133 & 8 \\ 3530 & 4037 \end{array}\right),\left(\begin{array}{rr} 9 & 16 \\ 194 & 3405 \end{array}\right)$.
The torsion field $K:=\Q(E[4080])$ is a degree-$57755566080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4080\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 20400bt
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 510e5, 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 |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.3421020160000.9 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.16.420297610421796509924667791769600000000.7 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | 16.0.26268600651362281870291736985600000000.34 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | add | nonsplit |
| $\lambda$-invariant(s) | - | 0 | - | 0 |
| $\mu$-invariant(s) | - | 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 $0$.