Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-6123x-2709322\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-6123xz^2-2709322z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6123x-2709322\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1379, 51100)$ | $4.6438180082969173026880501272$ | $\infty$ |
| $(154, 0)$ | $0$ | $2$ |
Integral points
\( \left(154, 0\right) \), \((1379,\pm 51100)\)
Invariants
| Conductor: | $N$ | = | \( 6120 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-3156372178560000$ | = | $-1 \cdot 2^{10} \cdot 3^{10} \cdot 5^{4} \cdot 17^{4} $ |
|
| j-invariant: | $j$ | = | \( -\frac{34008619684}{4228250625} \) | = | $-1 \cdot 2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 17^{-4} \cdot 157^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6534703974039053951409594292$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52654160260322945826231004286$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0094176599676135$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.94850246108947$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6438180082969173026880501272$ |
|
| Real period: | $\Omega$ | ≈ | $0.19911202667913608149392194255$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.6985600606442735160734016786 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.698560061 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.199112 \cdot 4.643818 \cdot 16}{2^2} \\ & \approx 3.698560061\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24576 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 4.24.0.2 |
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 $3$, and generators
$\left(\begin{array}{rr} 4065 & 16 \\ 4064 & 17 \end{array}\right),\left(\begin{array}{rr} 1456 & 4011 \\ 2835 & 376 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 817 & 2736 \\ 1644 & 193 \end{array}\right),\left(\begin{array}{rr} 2719 & 0 \\ 0 & 4079 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 241 & 2736 \\ 2892 & 193 \end{array}\right),\left(\begin{array}{rr} 3073 & 852 \\ 960 & 1717 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 680 = 2^{3} \cdot 5 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 6120.h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040.f4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.277102632960000.78 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.17318914560000.89 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.9469116535680000.5 | \(\Z/6\Z\) | 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 |
| $16$ | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | nonsplit | ss | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 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.