Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-443525x+114129789\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-443525xz^2+114129789z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-7096395x+7297210118\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(387, 384\right) \) | $1.2212519836475062775262869112$ | $\infty$ |
| \( \left(-769, 384\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([387:384:1]\) | $1.2212519836475062775262869112$ | $\infty$ |
| \([-769:384:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1547, 4624\right) \) | $1.2212519836475062775262869112$ | $\infty$ |
| \( \left(-3077, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-769, 384\right) \), \( \left(-753, 4928\right) \), \( \left(-753, -4176\right) \), \( \left(383, 384\right) \), \( \left(383, -768\right) \), \( \left(387, 384\right) \), \( \left(387, -772\right) \), \( \left(82463, 23638272\right) \), \( \left(82463, -23720736\right) \)
\([-769:384:1]\), \([-753:4928:1]\), \([-753:-4176:1]\), \([383:384:1]\), \([383:-768:1]\), \([387:384:1]\), \([387:-772:1]\), \([82463:23638272:1]\), \([82463:-23720736:1]\)
\( \left(-3077, 0\right) \), \((-3013,\pm 36416)\), \((1531,\pm 4608)\), \((1547,\pm 4624)\), \((329851,\pm 189436032)\)
Invariants
| Conductor: | $N$ | = | \( 36414 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 17^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-32289328885137408$ | = | $-1 \cdot 2^{18} \cdot 3^{6} \cdot 7 \cdot 17^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{548347731625}{1835008} \) | = | $-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0324402065052746886945562215$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.066527390143111802872166294100$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0293250379726038$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.8203656024081$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2212519836475062775262869112$ |
|
| Real period: | $\Omega$ | ≈ | $0.37121179689619667577623150122$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ ( 2 \cdot 3^{2} )\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $8.1601765796310372761729608821 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 8.160176580 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.371212 \cdot 1.221252 \cdot 72}{2^2} \\ & \approx 8.160176580\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 331776 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 8.6.0.1 | $6$ |
| $3$ | 3B | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2143 & 3060 \\ 2278 & 4897 \end{array}\right),\left(\begin{array}{rr} 8533 & 36 \\ 8532 & 37 \end{array}\right),\left(\begin{array}{rr} 611 & 5508 \\ 3944 & 475 \end{array}\right),\left(\begin{array}{rr} 3023 & 0 \\ 0 & 8567 \end{array}\right),\left(\begin{array}{rr} 4285 & 3060 \\ 6562 & 4897 \end{array}\right),\left(\begin{array}{rr} 3316 & 5049 \\ 3247 & 2194 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[8568])$ is a degree-$1091580198912$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8568\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 18207 = 3^{2} \cdot 7 \cdot 17^{2} \) |
| $3$ | additive | $6$ | \( 2023 = 7 \cdot 17^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 36414cf
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a3, its twist by $-51$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{51 +102 \sqrt{2}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.25798099131.3 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.77394297393.1 | \(\Z/18\Z\) | not in database |
| $8$ | 8.0.815021191627776.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.66532342173696.118 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1357802901504.8 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.22452684473388229641.2 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.293503986178850203226001.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | split | add | ss | nonsplit | ss | ord | add | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 10 | - | 1,3 | 1 | 1,3 | 1 | - | 1 | 1,1 | 1 | 1 | 1 | 1 | 3 | 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.