Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-128617x-17502779\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-128617xz^2-17502779z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-166688307x-814109335794\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-198, 599\right) \) | $1.0459897774500802664424714566$ | $\infty$ |
| \( \left(-\frac{917}{4}, \frac{917}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-198:599:1]\) | $1.0459897774500802664424714566$ | $\infty$ |
| \([-1834:917:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-7113, 108000\right) \) | $1.0459897774500802664424714566$ | $\infty$ |
| \( \left(-8238, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-198, 599\right) \), \( \left(-198, -401\right) \), \( \left(527, 7574\right) \), \( \left(527, -8101\right) \), \( \left(682, 14239\right) \), \( \left(682, -14921\right) \)
\([-198:599:1]\), \([-198:-401:1]\), \([527:7574:1]\), \([527:-8101:1]\), \([682:14239:1]\), \([682:-14921:1]\)
\((-7113,\pm 108000)\), \((18987,\pm 1692900)\), \((24567,\pm 3149280)\)
Invariants
| Conductor: | $N$ | = | \( 10470 \) | = | $2 \cdot 3 \cdot 5 \cdot 349$ |
|
| Minimal Discriminant: | $\Delta$ | = | $4636822725000000$ | = | $2^{6} \cdot 3^{12} \cdot 5^{8} \cdot 349 $ |
|
| j-invariant: | $j$ | = | \( \frac{235301185027835613721}{4636822725000000} \) | = | $2^{-6} \cdot 3^{-12} \cdot 5^{-8} \cdot 349^{-1} \cdot 6173641^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7981276278555753094683034823$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7981276278555753094683034823$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9705960517308437$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.067635401043438$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0459897774500802664424714566$ |
|
| Real period: | $\Omega$ | ≈ | $0.25240301546391679120536362419$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2^{3}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.1120877917826519394426941692 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.112087792 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.252403 \cdot 1.045990 \cdot 32}{2^2} \\ & \approx 2.112087792\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 124416 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $349$ | $1$ | $I_{1}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.12.0.8 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8376 = 2^{3} \cdot 3 \cdot 349 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5585 & 8 \\ 5588 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 8370 & 8371 \end{array}\right),\left(\begin{array}{rr} 8369 & 8 \\ 8368 & 9 \end{array}\right),\left(\begin{array}{rr} 5600 & 3 \\ 2453 & 2 \end{array}\right),\left(\begin{array}{rr} 5243 & 5236 \\ 5282 & 1053 \end{array}\right),\left(\begin{array}{rr} 1051 & 1050 \\ 3154 & 7339 \end{array}\right)$.
The torsion field $K:=\Q(E[8376])$ is a degree-$22721823129600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8376\Z)$.
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 | $4$ | \( 349 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1745 = 5 \cdot 349 \) |
| $5$ | split multiplicative | $6$ | \( 2094 = 2 \cdot 3 \cdot 349 \) |
| $349$ | split multiplicative | $350$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 10470.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{349}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-349}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt[4]{1396})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{349})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.118422027507164839936.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.40410648576.1 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 349 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.