Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+943197x-1667423198\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+943197xz^2-1667423198z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+943197x-1667423198\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11414471/4225, 39415979394/274625)$ | $14.725002056285826207422398248$ | $\infty$ |
| $(926, 0)$ | $0$ | $2$ |
Integral points
\( \left(926, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 12240 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-1254791250000000000000$ | = | $-1 \cdot 2^{13} \cdot 3^{10} \cdot 5^{16} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{31077313442863199}{420227050781250} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{-16} \cdot 17^{-1} \cdot 314399^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7291932018071368000065097491$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4867398769131366448916550092$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.042914731101016$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.94873274232582$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $14.725002056285826207422398248$ |
|
| Real period: | $\Omega$ | ≈ | $0.075091156925302534784425247549$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.4228697605358459348703706636 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.422869761 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.075091 \cdot 14.725002 \cdot 16}{2^2} \\ & \approx 4.422869761\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 393216 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 16.48.0.144 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8160 = 2^{5} \cdot 3 \cdot 5 \cdot 17 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 8129 & 32 \\ 8128 & 33 \end{array}\right),\left(\begin{array}{rr} 3074 & 31 \\ 1949 & 5700 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 5598 & 6155 \end{array}\right),\left(\begin{array}{rr} 5417 & 8134 \\ 2538 & 2869 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 7620 & 8129 \\ 1681 & 156 \end{array}\right),\left(\begin{array}{rr} 4897 & 32 \\ 4912 & 513 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3856 & 25 \\ 5879 & 3186 \end{array}\right)$.
The torsion field $K:=\Q(E[8160])$ is a degree-$924089057280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8160\Z)$.
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 | $4$ | \( 153 = 3^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 1360 = 2^{4} \cdot 5 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 12240.o
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 510.e8, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-34}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{102}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{34})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.32801857915060224.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.32801857915060224.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.443364212736.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.6136528896.5 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.8200464478765056.110 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.8200464478765056.105 | \(\Z/16\Z\) | 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 |
| $16$ | 16.0.1075961882679799065407149546930176.11 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | \(\Z/12\Z\) | 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 | ord | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 1 | 1,1 | 1 | 1 | 1 | 3 | 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 | 0,0 |
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.