Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-3963x+166417\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-3963xz^2+166417z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5136075x+7779759750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(222, 3089\right) \) | $0.060736309405755277074399474545$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([222:3089:1]\) | $0.060736309405755277074399474545$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7995, 691200\right) \) | $0.060736309405755277074399474545$ | $\infty$ |
Integral points
\( \left(-78, 89\right) \), \( \left(-78, -11\right) \), \( \left(-34, 529\right) \), \( \left(-34, -495\right) \), \( \left(22, 289\right) \), \( \left(22, -311\right) \), \( \left(62, 369\right) \), \( \left(62, -431\right) \), \( \left(68, 427\right) \), \( \left(68, -495\right) \), \( \left(222, 3089\right) \), \( \left(222, -3311\right) \), \( \left(20462, 2916769\right) \), \( \left(20462, -2937231\right) \)
\([-78:89:1]\), \([-78:-11:1]\), \([-34:529:1]\), \([-34:-495:1]\), \([22:289:1]\), \([22:-311:1]\), \([62:369:1]\), \([62:-431:1]\), \([68:427:1]\), \([68:-495:1]\), \([222:3089:1]\), \([222:-3311:1]\), \([20462:2916769:1]\), \([20462:-2937231:1]\)
\((-2805,\pm 10800)\), \((-1221,\pm 110592)\), \((795,\pm 64800)\), \((2235,\pm 86400)\), \((2451,\pm 99576)\), \((7995,\pm 691200)\), \((736635,\pm 632232000)\)
Invariants
| Conductor: | $N$ | = | \( 2450 \) | = | $2 \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-8028160000000$ | = | $-1 \cdot 2^{21} \cdot 5^{7} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{8990558521}{10485760} \) | = | $-1 \cdot 2^{-21} \cdot 5^{-1} \cdot 7 \cdot 1087^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1702447627977011860774865647$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.041207448404765447926214774180$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9865830459726923$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.814501398077582$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.060736309405755277074399474545$ |
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| Real period: | $\Omega$ | ≈ | $0.66883567156707739251963045758$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 84 $ = $ ( 3 \cdot 7 )\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4122992643519469177259602355 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.412299264 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.668836 \cdot 0.060736 \cdot 84}{1^2} \\ & \approx 3.412299264\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6048 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
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 |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 631 & 6 \\ 213 & 19 \end{array}\right),\left(\begin{array}{rr} 835 & 6 \\ 834 & 7 \end{array}\right),\left(\begin{array}{rr} 503 & 834 \\ 669 & 821 \end{array}\right),\left(\begin{array}{rr} 421 & 6 \\ 423 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 723 & 2 \\ 730 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 458 & 387 \\ 535 & 427 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$4459069440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | good | $2$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 98 = 2 \cdot 7^{2} \) |
| $7$ | additive | $14$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 2450.v
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 490.d1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.1960.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.153664000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.38288446875.5 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.3630312000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.181014965550212124968628705000000000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.14714420813917563377016000000000000000.2 | \(\Z/6\Z\) | 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 | split | ord | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 5 | - | - | 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 |
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.