Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-18491360x-218306963325\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-18491360xz^2-218306963325z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-295861755x-13971941514538\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7037, -3519)$ | $0$ | $2$ |
Integral points
\( \left(7037, -3519\right) \)
Invariants
| Conductor: | $N$ | = | \( 20286 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-20184441325492274617909248$ | = | $-1 \cdot 2^{30} \cdot 3^{10} \cdot 7^{12} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( -\frac{8152944444844179625}{235342826399858688} \) | = | $-1 \cdot 2^{-30} \cdot 3^{-4} \cdot 5^{3} \cdot 7^{-6} \cdot 23^{-1} \cdot 383^{3} \cdot 1051^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5360495275350938411052274612$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0137883086733823428549284710$ |
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| $abc$ quality: | $Q$ | ≈ | $1.052056363650569$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.6287158196198$ | |||
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: | $\Omega$ | ≈ | $0.029697944366903376573237671443$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 240 $ = $ ( 2 \cdot 3 \cdot 5 )\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7818766620142025943942602866 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.781876662 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.029698 \cdot 1.000000 \cdot 240}{2^2} \\ & \approx 1.781876662\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5529600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $30$ | $I_{30}$ | split multiplicative | -1 | 1 | 30 | 30 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $23$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | 8.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2575 & 1092 \\ 644 & 3863 \end{array}\right),\left(\begin{array}{rr} 1275 & 2464 \\ 686 & 3557 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2626 & 1659 \\ 693 & 2752 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3814 & 3855 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 1933 & 2772 \\ 3318 & 1177 \end{array}\right),\left(\begin{array}{rr} 3853 & 12 \\ 3852 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$413653008384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $4$ | \( 10143 = 3^{2} \cdot 7^{2} \cdot 23 \) |
| $3$ | additive | $8$ | \( 1127 = 7^{2} \cdot 23 \) |
| $5$ | good | $2$ | \( 10143 = 3^{2} \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $32$ | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 20286.bz
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 966.f3, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.649152.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.17003536814061.3 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.222919710806016.163 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.421398319104.41 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.313538501162411099906738100476514680696832.2 | \(\Z/18\Z\) | 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 | 5 | 7 | 23 |
|---|---|---|---|---|---|
| Reduction type | split | add | ss | add | nonsplit |
| $\lambda$-invariant(s) | 6 | - | 2,2 | - | 0 |
| $\mu$-invariant(s) | 0 | - | 0,0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.