Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-1330145x+291422025\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-1330145xz^2+291422025z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1723867947x+13601757602214\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(190, 6655\right) \) | $1.3758902091257469581614346167$ | $\infty$ |
| \( \left(1090, 11155\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([190:6655:1]\) | $1.3758902091257469581614346167$ | $\infty$ |
| \([1090:11155:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6843, 1458000\right) \) | $1.3758902091257469581614346167$ | $\infty$ |
| \( \left(39243, 2527200\right) \) | $0$ | $7$ |
Integral points
\( \left(-1250, 1795\right) \), \( \left(-1250, -545\right) \), \( \left(-860, 28705\right) \), \( \left(-860, -27845\right) \), \( \left(-158, 22387\right) \), \( \left(-158, -22229\right) \), \( \left(180, 7515\right) \), \( \left(180, -7695\right) \), \( \left(190, 6655\right) \), \( \left(190, -6845\right) \), \( \left(1090, 11155\right) \), \( \left(1090, -12245\right) \), \( \left(1870, 64975\right) \), \( \left(1870, -66845\right) \), \( \left(6940, 566905\right) \), \( \left(6940, -573845\right) \), \( \left(7810, 678955\right) \), \( \left(7810, -686765\right) \), \( \left(2060290, 2956250155\right) \), \( \left(2060290, -2958310445\right) \)
\([-1250:1795:1]\), \([-1250:-545:1]\), \([-860:28705:1]\), \([-860:-27845:1]\), \([-158:22387:1]\), \([-158:-22229:1]\), \([180:7515:1]\), \([180:-7695:1]\), \([190:6655:1]\), \([190:-6845:1]\), \([1090:11155:1]\), \([1090:-12245:1]\), \([1870:64975:1]\), \([1870:-66845:1]\), \([6940:566905:1]\), \([6940:-573845:1]\), \([7810:678955:1]\), \([7810:-686765:1]\), \([2060290:2956250155:1]\), \([2060290:-2958310445:1]\)
\((-44997,\pm 252720)\), \((-30957,\pm 6107400)\), \((-5685,\pm 4818528)\), \((6483,\pm 1642680)\), \((6843,\pm 1458000)\), \((39243,\pm 2527200)\), \((67323,\pm 14236560)\), \((249843,\pm 123201000)\), \((281163,\pm 147497760)\), \((74170443,\pm 638772544800)\)
Invariants
| Conductor: | $N$ | = | \( 32370 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 83$ |
|
| Minimal Discriminant: | $\Delta$ | = | $113901735543570000000$ | = | $2^{7} \cdot 3^{7} \cdot 5^{7} \cdot 13^{7} \cdot 83 $ |
|
| j-invariant: | $j$ | = | \( \frac{260267950003303480801681}{113901735543570000000} \) | = | $2^{-7} \cdot 3^{-7} \cdot 5^{-7} \cdot 13^{-7} \cdot 83^{-1} \cdot 199^{3} \cdot 320839^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5444499742326380300736641427$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5444499742326380300736641427$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.987305727649173$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.191725018917374$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3758902091257469581614346167$ |
|
| Real period: | $\Omega$ | ≈ | $0.16851155954991018867834113670$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2401 $ = $ 7\cdot7\cdot7\cdot7\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $11.360816840552352881555330864 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 11.360816841 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.168512 \cdot 1.375890 \cdot 2401}{7^2} \\ & \approx 11.360816841\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 998816 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $83$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 906360 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 83 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 226591 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 453181 & 14 \\ 453187 & 99 \end{array}\right),\left(\begin{array}{rr} 578761 & 14 \\ 425887 & 99 \end{array}\right),\left(\begin{array}{rr} 543817 & 14 \\ 181279 & 99 \end{array}\right),\left(\begin{array}{rr} 679771 & 453194 \\ 0 & 32371 \end{array}\right),\left(\begin{array}{rr} 906347 & 14 \\ 906346 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 69721 & 14 \\ 488047 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 604241 & 14 \\ 604247 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[906360])$ is a degree-$913086556114004213760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/906360\Z)$.
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 | $4$ | \( 16185 = 3 \cdot 5 \cdot 13 \cdot 83 \) |
| $3$ | split multiplicative | $4$ | \( 10790 = 2 \cdot 5 \cdot 13 \cdot 83 \) |
| $5$ | split multiplicative | $6$ | \( 6474 = 2 \cdot 3 \cdot 13 \cdot 83 \) |
| $7$ | good | $2$ | \( 83 \) |
| $13$ | split multiplicative | $14$ | \( 2490 = 2 \cdot 3 \cdot 5 \cdot 83 \) |
| $83$ | nonsplit multiplicative | $84$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 32370bk
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.129480.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.6.2170741315392000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | 83 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | split | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 2 | 8 | 2 | 5 | 1 | 4 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.