Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-4765x+127781\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-4765xz^2+127781z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-76235x+8101766\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(39, -8)$ | $0.48084412869197135305267508211$ | $\infty$ |
Integral points
\( \left(39, -8\right) \), \( \left(39, -32\right) \)
Invariants
| Conductor: | $N$ | = | \( 12482 \) | = | $2 \cdot 79^{2}$ |
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| Discriminant: | $\Delta$ | = | $798848$ | = | $2^{7} \cdot 79^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1916782322625}{128} \) | = | $2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 79 \cdot 193^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.58910760721771544366853302939$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13913370152678813869362456086$ |
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| $abc$ quality: | $Q$ | ≈ | $1.046380923886781$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.924978414205276$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.48084412869197135305267508211$ |
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| Real period: | $\Omega$ | ≈ | $2.1427755107265412757159478680$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 7 $ = $ 7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.2123871640645835893625692867 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.212387164 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.142776 \cdot 0.480844 \cdot 7}{1^2} \\ & \approx 7.212387164\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5460 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 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 |
| $79$ | $1$ | $II$ | additive | -1 | 2 | 2 | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.2 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4424 = 2^{3} \cdot 7 \cdot 79 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 3643 & 8 \\ 2170 & 4387 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4411 & 14 \\ 4410 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 3321 & 640 \\ 2198 & 2017 \end{array}\right),\left(\begin{array}{rr} 2213 & 14 \\ 2219 & 99 \end{array}\right),\left(\begin{array}{rr} 3319 & 14 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4424])$ is a degree-$1240271585280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4424\Z)$.
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 | $4$ | \( 6241 = 79^{2} \) |
| $7$ | good | $2$ | \( 6241 = 79^{2} \) |
| $79$ | additive | $1094$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 12482f
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.49928.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.19942441472.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.51716086897993.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.18.36259089263769930839653227889307494647242948608.1 | \(\Z/14\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ss | ss | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ss | add |
| $\lambda$-invariant(s) | 2 | 9,1 | 1,1 | 1 | 1 | 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 | 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.