Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-142x+701\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-142xz^2+701z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-11529x+476469\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 1)$ | $0.25810635289973679058716498246$ | $\infty$ |
Integral points
\((-1,\pm 29)\), \((7,\pm 1)\)
Invariants
| Conductor: | $N$ | = | \( 196 \) | = | $2^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $784$ | = | $2^{4} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( 406749952 \) | = | $2^{8} \cdot 7 \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.20503477820738141011106300958$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.76040219656991539743436584064$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9989714715999977$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.018481013875294$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.25810635289973679058716498246$ |
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| Real period: | $\Omega$ | ≈ | $4.3609185288243093391936038281$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.1255807767677281735952769161 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.125580777 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.360919 \cdot 0.258106 \cdot 1}{1^2} \\ & \approx 1.125580777\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $7$ | $1$ | $II$ | 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 |
|---|---|---|
| $2$ | 2Cn | 2.2.0.1 |
| $3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \), index $864$, genus $28$, and generators
$\left(\begin{array}{rr} 146 & 9 \\ 45 & 131 \end{array}\right),\left(\begin{array}{rr} 163 & 36 \\ 114 & 47 \end{array}\right),\left(\begin{array}{rr} 217 & 36 \\ 216 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 36 \\ 18 & 145 \end{array}\right),\left(\begin{array}{rr} 31 & 6 \\ 112 & 103 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 42 & 199 \end{array}\right)$.
The torsion field $K:=\Q(E[252])$ is a degree-$870912$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/252\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $14$ | \( 4 = 2^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 196.a
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/3\Z\) | 2.2.21.1-784.1-d2 |
| $3$ | \(\Q(\zeta_{7})^+\) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.2420208.1 | \(\Z/3\Z\) | not in database |
| $6$ | \(\Q(\zeta_{21})^+\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.4.74049191673856.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $12$ | 12.0.52716660869376.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $18$ | 18.6.148287459778733772774300635136.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.14176142707705638912.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.