Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+597439x+378137535\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+597439xz^2+378137535z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+48392532x+275517085392\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(466, 27531)$ | $4.4222341958990168981975758848$ | $\infty$ |
| $(-465, 0)$ | $0$ | $2$ |
Integral points
\( \left(-465, 0\right) \), \((466,\pm 27531)\)
Invariants
| Conductor: | $N$ | = | \( 18240 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-75353483221401600000$ | = | $-1 \cdot 2^{38} \cdot 3^{5} \cdot 5^{5} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{89962967236397039}{287450726400000} \) | = | $2^{-20} \cdot 3^{-5} \cdot 5^{-5} \cdot 19^{-2} \cdot 29^{3} \cdot 15451^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4972573453549331039271061505$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4575365745150151398012579683$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0281324592856305$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.404303048338218$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.4222341958990168981975758848$ |
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| Real period: | $\Omega$ | ≈ | $0.13687430927258214852718282594$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2^{2}\cdot5\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.0529025100527067012864928226 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.052902510 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.136874 \cdot 4.422234 \cdot 40}{2^2} \\ & \approx 6.052902510\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 460800 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{28}^{*}$ | additive | -1 | 6 | 38 | 20 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $19$ | $2$ | $I_{2}$ | nonsplit 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 | 2.3.0.1 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 571 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2274 & 2267 \\ 1985 & 1324 \end{array}\right),\left(\begin{array}{rr} 246 & 13 \\ 155 & 112 \end{array}\right),\left(\begin{array}{rr} 2261 & 20 \\ 2260 & 21 \end{array}\right),\left(\begin{array}{rr} 1139 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 2269 & 2264 \\ 1380 & 1489 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 2040 & 1931 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1499 & 2274 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1159 & 2260 \\ 1160 & 2259 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$15128985600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 15 = 3 \cdot 5 \) |
| $3$ | split multiplicative | $4$ | \( 6080 = 2^{6} \cdot 5 \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1216 = 2^{6} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 18240.cc
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 570.l4, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/10\Z\) | not in database |
| $4$ | 4.2.1386240.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.0.16842816000000.88 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.432373800960000.69 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1921661337600.26 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.9451648239783810422571008000000000000000.1 | \(\Z/10\Z\) | 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 | add | split | nonsplit | ord | ord | ord | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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.