Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-2403008x+1434568512\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-2403008xz^2+1434568512z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-194643675x+1045216514250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(872, 1200\right) \) | $1.5894226529454628204885410764$ | $\infty$ |
| \( \left(802, 4750\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([872:1200:1]\) | $1.5894226529454628204885410764$ | $\infty$ |
| \([802:4750:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7845, 32400\right) \) | $1.5894226529454628204885410764$ | $\infty$ |
| \( \left(7215, 128250\right) \) | $0$ | $4$ |
Integral points
\((536,\pm 17328)\), \((802,\pm 4750)\), \((872,\pm 1200)\), \( \left(897, 0\right) \), \((5552,\pm 399000)\)
\([536:\pm 17328:1]\), \([802:\pm 4750:1]\), \([872:\pm 1200:1]\), \([897:0:1]\), \([5552:\pm 399000:1]\)
\((536,\pm 17328)\), \((802,\pm 4750)\), \((872,\pm 1200)\), \( \left(897, 0\right) \), \((5552,\pm 399000)\)
Invariants
| Conductor: | $N$ | = | \( 22800 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $9383112000000000$ | = | $2^{12} \cdot 3^{2} \cdot 5^{9} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{23977812996389881}{146611125} \) | = | $3^{-2} \cdot 5^{-3} \cdot 19^{-4} \cdot 288361^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2520049916038494961499279491$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.75413885482685399943231616103$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0982581996716574$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.549874055467068$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5894226529454628204885410764$ |
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| Real period: | $\Omega$ | ≈ | $0.36499121742132394184250200535$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.6410024727647594149689165447 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.641002473 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.364991 \cdot 1.589423 \cdot 128}{4^2} \\ & \approx 4.641002473\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 442368 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $19$ | $4$ | $I_{4}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.12.0.7 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 760 = 2^{3} \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 99 & 98 \\ 298 & 675 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 753 & 8 \\ 752 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 754 & 755 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 144 & 757 \\ 755 & 758 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 277 & 284 \\ 238 & 659 \end{array}\right),\left(\begin{array}{rr} 401 & 8 \\ 84 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[760])$ is a degree-$1891123200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/760\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 22800ch
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 285c3, its twist by $-20$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15 + \sqrt{-95}})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.5184000000.15 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.8340544000000.23 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.432373800960000.11 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | add | ord | ord | ord | ord | split | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 1 | 1 | 3 | 1 | 2 | 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 |
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.