Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+11690x+2913572\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+11690xz^2+2913572z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+15150213x+135890164566\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-76, 1298\right) \) | $0.076367827046566519937277182316$ | $\infty$ |
| \( \left(-116, 58\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-76:1298:1]\) | $0.076367827046566519937277182316$ | $\infty$ |
| \([-116:58:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2733, 272160\right) \) | $0.076367827046566519937277182316$ | $\infty$ |
| \( \left(-4173, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-116, 58\right) \), \( \left(-112, 506\right) \), \( \left(-112, -394\right) \), \( \left(-76, 1298\right) \), \( \left(-76, -1222\right) \), \( \left(-16, 1658\right) \), \( \left(-16, -1642\right) \), \( \left(14, 1748\right) \), \( \left(14, -1762\right) \), \( \left(92, 2138\right) \), \( \left(92, -2230\right) \), \( \left(134, 2558\right) \), \( \left(134, -2692\right) \), \( \left(284, 5258\right) \), \( \left(284, -5542\right) \), \( \left(404, 8378\right) \), \( \left(404, -8782\right) \), \( \left(694, 18238\right) \), \( \left(694, -18932\right) \), \( \left(1184, 40358\right) \), \( \left(1184, -41542\right) \), \( \left(2444, 119738\right) \), \( \left(2444, -122182\right) \), \( \left(12884, 1456058\right) \), \( \left(12884, -1468942\right) \)
\([-116:58:1]\), \([-112:506:1]\), \([-112:-394:1]\), \([-76:1298:1]\), \([-76:-1222:1]\), \([-16:1658:1]\), \([-16:-1642:1]\), \([14:1748:1]\), \([14:-1762:1]\), \([92:2138:1]\), \([92:-2230:1]\), \([134:2558:1]\), \([134:-2692:1]\), \([284:5258:1]\), \([284:-5542:1]\), \([404:8378:1]\), \([404:-8782:1]\), \([694:18238:1]\), \([694:-18932:1]\), \([1184:40358:1]\), \([1184:-41542:1]\), \([2444:119738:1]\), \([2444:-122182:1]\), \([12884:1456058:1]\), \([12884:-1468942:1]\)
\( \left(-4173, 0\right) \), \((-4029,\pm 97200)\), \((-2733,\pm 272160)\), \((-573,\pm 356400)\), \((507,\pm 379080)\), \((3315,\pm 471744)\), \((4827,\pm 567000)\), \((10227,\pm 1166400)\), \((14547,\pm 1853280)\), \((24987,\pm 4014360)\), \((42627,\pm 8845200)\), \((87987,\pm 26127360)\), \((463827,\pm 315900000)\)
Invariants
| Conductor: | $N$ | = | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-3766993776000000$ | = | $-1 \cdot 2^{10} \cdot 3^{7} \cdot 5^{6} \cdot 7^{2} \cdot 13^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{80414592731747}{1714608000000} \) | = | $2^{-10} \cdot 3^{-7} \cdot 5^{-6} \cdot 7^{-2} \cdot 17^{3} \cdot 2539^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6690603017549793120718428672$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0278229623895951280584710068$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0149254815360271$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.1321209950302995$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.076367827046566519937277182316$ |
|
| Real period: | $\Omega$ | ≈ | $0.33077779462668284197734331565$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1680 $ = $ ( 2 \cdot 5 )\cdot7\cdot( 2 \cdot 3 )\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $10.609528192575990563880160336 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 10.609528193 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.330778 \cdot 0.076368 \cdot 1680}{2^2} \\ & \approx 10.609528193\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 241920 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1204 & 1 \\ 479 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1042 & 1 \\ 1039 & 0 \end{array}\right),\left(\begin{array}{rr} 1557 & 4 \\ 1556 & 5 \end{array}\right),\left(\begin{array}{rr} 977 & 586 \\ 584 & 975 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 781 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 937 & 4 \\ 314 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 39 = 3 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 3549 = 3 \cdot 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 35490dr
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.10545600.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1000887114240000.79 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
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 | nonsplit | ord | add | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 4 | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.