Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-64175x-163709355\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-64175xz^2-163709355z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-83170827x-7637774154426\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1762, 71191\right) \) | $6.7680636504127631689851903115$ | $\infty$ |
| \( \left(586, -293\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1762:71191:1]\) | $6.7680636504127631689851903115$ | $\infty$ |
| \([586:-293:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(63435, 15567552\right) \) | $6.7680636504127631689851903115$ | $\infty$ |
| \( \left(21099, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(586, -293\right) \), \( \left(1762, 71191\right) \), \( \left(1762, -72953\right) \)
\([586:-293:1]\), \([1762:71191:1]\), \([1762:-72953:1]\)
\( \left(21099, 0\right) \), \((63435,\pm 15567552)\)
Invariants
| Conductor: | $N$ | = | \( 6270 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 19$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-11560253601080069820$ | = | $-1 \cdot 2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{10} \cdot 19^{5} $ |
|
| j-invariant: | $j$ | = | \( -\frac{29229525625065721201}{11560253601080069820} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-2} \cdot 5^{-1} \cdot 11^{-10} \cdot 19^{-5} \cdot 3080401^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3369816556895193572319759973$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3369816556895193572319759973$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.040337758478602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.87294570772896$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.7680636504127631689851903115$ |
|
| Real period: | $\Omega$ | ≈ | $0.10156529336106537036482481273$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot2\cdot1\cdot( 2 \cdot 5 )\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.8740037014053527078658190679 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.874003701 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.101565 \cdot 6.768064 \cdot 40}{2^2} \\ & \approx 6.874003701\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 160000 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $19$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 | 2.3.0.1 | $3$ |
| $5$ | 5B.1.2 | 5.24.0.3 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25080 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 25061 & 20 \\ 25060 & 21 \end{array}\right),\left(\begin{array}{rr} 16721 & 20 \\ 16730 & 201 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 24840 & 24731 \end{array}\right),\left(\begin{array}{rr} 9121 & 20 \\ 15970 & 201 \end{array}\right),\left(\begin{array}{rr} 5032 & 5 \\ 25035 & 25066 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 6271 & 20 \\ 12550 & 201 \end{array}\right),\left(\begin{array}{rr} 7936 & 5 \\ 18435 & 25066 \end{array}\right),\left(\begin{array}{rr} 12541 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[25080])$ is a degree-$199702609920000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25080\Z)$.
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 | $4$ | \( 95 = 5 \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 2090 = 2 \cdot 5 \cdot 11 \cdot 19 \) |
| $5$ | split multiplicative | $6$ | \( 6 = 2 \cdot 3 \) |
| $11$ | split multiplicative | $12$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 6270r
consists of 4 curves linked by isogenies of
degrees dividing 10.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-95}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{97 +12 \sqrt{66}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{5})\) | \(\Z/10\Z\) | not in database |
| $5$ | 5.1.2531250000.13 | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2036265625.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.0.79324636420898437500000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.5131569027900695800781250000000000000000.15 | \(\Z/5\Z \oplus \Z/10\Z\) | not in database |
| $20$ | 20.2.497631011454939064202060220000000000000000000000000000000000000.1 | \(\Z/20\Z\) | 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 | split | split | ord | split | ord | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 1 | 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.