Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+220x+2192\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+220xz^2+2192z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+284445x+97999902\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-8, 4)$ | $0$ | $2$ |
Integral points
\( \left(-8, 4\right) \)
Invariants
| Conductor: | $N$ | = | \( 98 \) | = | $2 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2582630848$ | = | $-1 \cdot 2^{6} \cdot 7^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{9938375}{21952} \) | = | $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.49017632033671360972722004737$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.48277875419094304282545632435$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9869508090989833$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.286410124875624$ | |||
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: | $\Omega$ | ≈ | $1.0019771957698767743050371648$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0019771957698767743050371648 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.001977196 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.001977 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 1.001977196\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 325 & 276 \\ 18 & 163 \end{array}\right),\left(\begin{array}{rr} 410 & 501 \\ 99 & 308 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 18 & 145 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 432 & 307 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 49 = 7^{2} \) |
| $3$ | good | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $32$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 98.a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a6, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.7.1-28.2-a7 |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/6\Z\) | 2.2.21.1-28.1-a3 |
| $4$ | 4.2.448.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-7})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.120472576.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.9834496.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.4.796594176.2 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.634562281237118976.5 | \(\Z/6\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.148287459778733772774300635136.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.1839302601739695422127.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 7 |
|---|---|---|---|
| Reduction type | nonsplit | ord | add |
| $\lambda$-invariant(s) | 2 | 0 | - |
| $\mu$-invariant(s) | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.