Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-994725579x-12060749479815\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-994725579xz^2-12060749479815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1289164350411x-562686990264984186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-18155, 131324)$ | $4.3366408545266255642500139041$ | $\infty$ |
| $(-18733, 9366)$ | $0$ | $2$ |
Integral points
\( \left(-18733, 9366\right) \), \( \left(-18155, 131324\right) \), \( \left(-18155, -113170\right) \), \( \left(81619, 21183638\right) \), \( \left(81619, -21265258\right) \)
Invariants
| Conductor: | $N$ | = | \( 84966 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $157517120697727379614728192$ | = | $2^{24} \cdot 3^{4} \cdot 7^{10} \cdot 17^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{38331145780597164097}{55468445663232} \) | = | $2^{-24} \cdot 3^{-4} \cdot 7^{-4} \cdot 17^{-1} \cdot 431^{3} \cdot 7823^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9289699450388644910864026196$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5394081984830997984089589389$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0214246831520672$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.499338629768534$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3366408545266255642500139041$ |
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| Real period: | $\Omega$ | ≈ | $0.026885088813652818943660142334$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ ( 2^{3} \cdot 3 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5963667772894519681476808109 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.596366777 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.026885 \cdot 4.336641 \cdot 192}{2^2} \\ & \approx 5.596366777\end{aligned}$$
Modular invariants
Modular form 84966.2.a.cr
For more coefficients, see the Downloads section to the right.
| Modular degree: | 53084160 |
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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 not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $17$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 16.48.0.102 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1904 = 2^{4} \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 1900 & 1901 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 1903 \end{array}\right),\left(\begin{array}{rr} 1336 & 1631 \\ 273 & 1350 \end{array}\right),\left(\begin{array}{rr} 113 & 560 \\ 1330 & 1471 \end{array}\right),\left(\begin{array}{rr} 1 & 560 \\ 1428 & 1429 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1889 & 16 \\ 1888 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1806 & 1891 \end{array}\right)$.
The torsion field $K:=\Q(E[1904])$ is a degree-$20214448128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1904\Z)$.
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 | $4$ | \( 14161 = 7^{2} \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 14161 = 7^{2} \cdot 17^{2} \) |
| $7$ | additive | $32$ | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 84966.cr
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714.f5, its twist by $-119$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-119}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.4.14836301611264.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.14836301611264.19 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.821386940416.21 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | nonsplit | ord | add | ord | ord | add | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 11 | 1 | 3 | - | 3 | 1 | - | 1 | 1 | 3 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.