Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-6800x+240000\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-6800xz^2+240000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8812827x+11223878454\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(40, 160\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([40:160:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1443, 38880\right) \) | $0$ | $7$ |
Integral points
\( \left(-50, 700\right) \), \( \left(-50, -650\right) \), \( \left(40, 160\right) \), \( \left(40, -200\right) \), \( \left(100, 700\right) \), \( \left(100, -800\right) \)
\([-50:700:1]\), \([-50:-650:1]\), \([40:160:1]\), \([40:-200:1]\), \([100:700:1]\), \([100:-800:1]\)
\((-1797,\pm 145800)\), \((1443,\pm 38880)\), \((3603,\pm 162000)\)
Invariants
| Conductor: | $N$ | = | \( 6690 \) | = | $2 \cdot 3 \cdot 5 \cdot 223$ |
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| Minimal Discriminant: | $\Delta$ | = | $-4877010000000$ | = | $-1 \cdot 2^{7} \cdot 3^{7} \cdot 5^{7} \cdot 223 $ |
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| j-invariant: | $j$ | = | \( -\frac{34773983355859201}{4877010000000} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 5^{-7} \cdot 19^{3} \cdot 41^{3} \cdot 223^{-1} \cdot 419^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1659436442403933624654841278$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1659436442403933624654841278$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9421514052132133$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.348665450111981$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.74461157165765281924699162949$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 343 $ = $ 7\cdot7\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.2122810016035697347289414064 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 5.212281002 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.744612 \cdot 1.000000 \cdot 343}{7^2} \\ & \approx 5.212281002\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19992 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | 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 |
| $223$ | $1$ | $I_{1}$ | 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 \( 187320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 223 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 28568 & 7 \\ 21833 & 187314 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 46823 & 187314 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 140491 & 93674 \\ 0 & 86971 \end{array}\right),\left(\begin{array}{rr} 187307 & 14 \\ 187306 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 124888 & 7 \\ 124873 & 187314 \end{array}\right),\left(\begin{array}{rr} 74936 & 7 \\ 74921 & 187314 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 93653 & 187314 \end{array}\right)$.
The torsion field $K:=\Q(E[187320])$ is a degree-$1829581732781752320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/187320\Z)$.
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 | $4$ | \( 3345 = 3 \cdot 5 \cdot 223 \) |
| $3$ | split multiplicative | $4$ | \( 2230 = 2 \cdot 5 \cdot 223 \) |
| $5$ | split multiplicative | $6$ | \( 1338 = 2 \cdot 3 \cdot 223 \) |
| $7$ | good | $2$ | \( 223 \) |
| $223$ | nonsplit multiplicative | $224$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 6690j
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 |
|---|---|---|---|
| $3$ | 3.1.26760.3 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.19162771776000.2 | \(\Z/2\Z \oplus \Z/14\Z\) | 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 | 223 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | ord | nonsplit |
| $\lambda$-invariant(s) | 4 | 1 | 1 | 4 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.