Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-9651x+349198\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-9651xz^2+349198z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-12507075x+16329714750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-113, 56)$ | $0$ | $2$ |
| $(47, -24)$ | $0$ | $2$ |
Integral points
\( \left(-113, 56\right) \), \( \left(47, -24\right) \)
Invariants
| Conductor: | $N$ | = | \( 21450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $4601025000000$ | = | $2^{6} \cdot 3^{2} \cdot 5^{8} \cdot 11^{2} \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{6361447449889}{294465600} \) | = | $2^{-6} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-2} \cdot 2647^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1911508663298932761216843671$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.38643191011284308882130470049$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8963574766727707$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9241974541777993$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.76460028739969807868174087983$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $3.0584011495987923147269635193 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 3.058401150 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.764600 \cdot 1.000000 \cdot 64}{4^2} \\ & \approx 3.058401150\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 55296 |
|
| $ \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 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | split 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 | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7923 & 2 \\ 11878 & 17159 \end{array}\right),\left(\begin{array}{rr} 8581 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 6243 & 2 \\ 1558 & 17159 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11441 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17157 & 4 \\ 17156 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 6863 & 17158 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 4293 & 4 \\ 8582 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$255058771968000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $4$ | \( 3575 = 5^{2} \cdot 11 \cdot 13 \) |
| $5$ | additive | $18$ | \( 858 = 2 \cdot 3 \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 21450.bm
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 4290.q2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{3}, \sqrt{715})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{286})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 11 | 13 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | split | add | split | split |
| $\lambda$-invariant(s) | 3 | 1 | - | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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 $0$.