Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-42093157x+104393523905\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-42093157xz^2+104393523905z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-54552731499x+4870747909506150\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-7486, 3743)$ | $0$ | $2$ |
| $(3490, -1745)$ | $0$ | $2$ |
Integral points
\( \left(-7486, 3743\right) \), \( \left(3490, -1745\right) \)
Invariants
| Conductor: | $N$ | = | \( 4998 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $64997422544797276838976$ | = | $2^{6} \cdot 3^{16} \cdot 7^{10} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{70108386184777836280897}{552468975892674624} \) | = | $2^{-6} \cdot 3^{-16} \cdot 7^{-4} \cdot 17^{-4} \cdot 41234113^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2055104535707017603788674321$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2325553790430451078261910604$ |
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| $abc$ quality: | $Q$ | ≈ | $1.07814467781823$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.547417135192263$ | |||
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: | $\Omega$ | ≈ | $0.11085006093280237332234992672$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 768 $ = $ ( 2 \cdot 3 )\cdot2^{4}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.3208029247745139194727964825 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 5.320802925 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.110850 \cdot 1.000000 \cdot 768}{4^2} \\ & \approx 5.320802925\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $17$ | $2$ | $I_{4}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.48.0.96 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 952 = 2^{3} \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 785 & 280 \\ 420 & 169 \end{array}\right),\left(\begin{array}{rr} 687 & 448 \\ 294 & 393 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 951 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 945 & 8 \\ 944 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 948 & 949 \end{array}\right),\left(\begin{array}{rr} 547 & 378 \\ 98 & 267 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[952])$ is a degree-$1263403008$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/952\Z)$.
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 | $4$ | \( 49 = 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 833 = 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 4998.bq
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714.f2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.10070523904.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.13142191046656.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.1740675142665963.10 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.1622647227216566419456.13 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.172717185506805128768782336.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 17 |
|---|---|---|---|---|
| Reduction type | split | split | add | nonsplit |
| $\lambda$-invariant(s) | 7 | 7 | - | 0 |
| $\mu$-invariant(s) | 0 | 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 $0$.