Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-2273580x-1318734320\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-2273580xz^2-1318734320z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-36377283x-84435373762\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3517/4, 3517/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 10710 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $239213790287962560$ | = | $2^{6} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 17^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{1782900110862842086081}{328139630024640} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 527207^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3373914711441328129930558549$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7880853268100779672954332364$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9994946520793547$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.983899336389663$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.12294950429244081727254672512$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot1\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.98359603433952653818037380093 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.983596034 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.122950 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.983596034\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 294912 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | 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 | 8.24.0.90 |
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 $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1352 & 4079 \\ 2641 & 4070 \end{array}\right),\left(\begin{array}{rr} 4065 & 16 \\ 4064 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3982 & 4067 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 16 \\ 1928 & 129 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 3324 & 3385 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 4076 & 4077 \end{array}\right),\left(\begin{array}{rr} 2456 & 1 \\ 3343 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3578 & 1031 \\ 427 & 1926 \end{array}\right)$.
The torsion field $K:=\Q(E[4080])$ is a degree-$231022264320$ 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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $8$ | \( 595 = 5 \cdot 7 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 10710f
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3570t4, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-10})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.7001316000000.25 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1792336896000000.156 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3317760000.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.22202489064616875.10 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | \(\Z/12\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | add | nonsplit | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 7 | - | 0 | 0 | 0 |
| $\mu$-invariant(s) | 1 | - | 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 $0$.