Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-651334x-202327012\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-651334xz^2-202327012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-844128891x-9437236685226\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1889/4, 1889/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 5658 \) | = | $2 \cdot 3 \cdot 23 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $9478135170529992$ | = | $2^{3} \cdot 3^{2} \cdot 23^{8} \cdot 41^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{30558708894178043523937}{9478135170529992} \) | = | $2^{-3} \cdot 3^{-2} \cdot 23^{-8} \cdot 41^{-2} \cdot 31264033^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0419812988819549728710664416$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0419812988819549728710664416$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9997766120882076$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.991781127735375$ | |||
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: | $\Omega$ | ≈ | $0.16805703518871611241631711823$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.0333688445291866979916108374 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.033368845 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.168057 \cdot 1.000000 \cdot 24}{2^2} \\ & \approx 4.033368845\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 79872 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $23$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $41$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 8.48.0.173 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15088 = 2^{4} \cdot 23 \cdot 41 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15073 & 16 \\ 15072 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5671 & 16 \\ 1630 & 11001 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 15084 & 15085 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 14990 & 15075 \end{array}\right),\left(\begin{array}{rr} 6269 & 16 \\ 3056 & 14773 \end{array}\right),\left(\begin{array}{rr} 5249 & 16 \\ 11816 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11324 & 11321 \\ 7547 & 7546 \end{array}\right)$.
The torsion field $K:=\Q(E[15088])$ is a degree-$94220963020800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15088\Z)$.
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 | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 943 = 23 \cdot 41 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 138 = 2 \cdot 3 \cdot 23 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 5658.h
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.30984192.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.18432.1 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{82})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{41})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3840080615571456.53 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1358954496.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.185189072896.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.140081432044425147.8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | \(\Z/12\Z\) | 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 | 23 | 41 |
|---|---|---|---|---|
| Reduction type | split | split | nonsplit | split |
| $\lambda$-invariant(s) | 3 | 5 | 0 | 1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.