Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-207834x-36553572\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-207834xz^2-36553572z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-269353539x-1701403155522\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4991, 348622)$ | $5.1995674235613216749806270499$ | $\infty$ |
| $(-1061/4, 1061/8)$ | $0$ | $2$ |
Integral points
\( \left(4991, 348622\right) \), \( \left(4991, -353613\right) \)
Invariants
| Conductor: | $N$ | = | \( 93786 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 29$ |
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| Discriminant: | $\Delta$ | = | $65903034676248$ | = | $2^{3} \cdot 3^{2} \cdot 7^{6} \cdot 11 \cdot 29^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{8438952173768857}{560166552} \) | = | $2^{-3} \cdot 3^{-2} \cdot 11^{-1} \cdot 13^{3} \cdot 29^{-4} \cdot 15661^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7074589745240654583728621550$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73450389999640880582018578328$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9620912002646477$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.222907069114411$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1995674235613216749806270499$ |
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| Real period: | $\Omega$ | ≈ | $0.22360003299202988324189202267$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.6504937898103813783673802935 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.650493790 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.223600 \cdot 5.199567 \cdot 16}{2^2} \\ & \approx 4.650493790\end{aligned}$$
Modular invariants
Modular form 93786.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17864 = 2^{3} \cdot 7 \cdot 11 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 15632 & 1603 \\ 13083 & 13112 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15401 & 12768 \\ 5460 & 15345 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7655 & 0 \\ 0 & 17863 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 960 & 15001 \\ 11795 & 11782 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 17858 & 17859 \end{array}\right),\left(\begin{array}{rr} 17857 & 8 \\ 17856 & 9 \end{array}\right),\left(\begin{array}{rr} 14624 & 2555 \\ 3717 & 12762 \end{array}\right)$.
The torsion field $K:=\Q(E[17864])$ is a degree-$580830953472000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17864\Z)$.
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 | $4$ | \( 539 = 7^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 15631 = 7^{2} \cdot 11 \cdot 29 \) |
| $7$ | additive | $26$ | \( 1914 = 2 \cdot 3 \cdot 11 \cdot 29 \) |
| $11$ | split multiplicative | $12$ | \( 8526 = 2 \cdot 3 \cdot 7^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 93786.q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1914.e1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-77}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{22})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/12\Z\) | 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 | nonsplit | nonsplit | ord | add | split | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 1 | 1 | - | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 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.