Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+39652116x-439825302704\)
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(homogenize, simplify) |
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\(y^2z=x^3+39652116xz^2-439825302704z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+39652116x-439825302704\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(106684538/529, 1102432960512/12167)$ | $9.6113365767905524486577504141$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 16704 \) | = | $2^{6} \cdot 3^{2} \cdot 29$ |
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| Discriminant: | $\Delta$ | = | $-87558857059662282627219456$ | = | $-1 \cdot 2^{51} \cdot 3^{13} \cdot 29^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{36079072622241241607}{458176313589497856} \) | = | $2^{-33} \cdot 3^{-7} \cdot 7^{3} \cdot 29^{-3} \cdot 103^{3} \cdot 4583^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6586894603413153002282368271$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0696625451673424904047660265$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0885358591274659$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.905239764385579$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.6113365767905524486577504141$ |
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| Real period: | $\Omega$ | ≈ | $0.029686064803097963679180188924$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.2825820837039207011354066727 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.282582084 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.029686 \cdot 9.611337 \cdot 8}{1^2} \\ & \approx 2.282582084\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7096320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{41}^{*}$ | additive | -1 | 6 | 51 | 33 |
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $29$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 696 = 2^{3} \cdot 3 \cdot 29 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 142 & 549 \\ 695 & 86 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 347 & 690 \\ 345 & 677 \end{array}\right),\left(\begin{array}{rr} 691 & 6 \\ 690 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 521 & 690 \\ 0 & 695 \end{array}\right),\left(\begin{array}{rr} 553 & 6 \\ 267 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[696])$ is a degree-$3143024640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/696\Z)$.
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 | $4$ | \( 261 = 3^{2} \cdot 29 \) |
| $3$ | additive | $8$ | \( 64 = 2^{6} \) |
| $29$ | nonsplit multiplicative | $30$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 16704cv
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 174a2, its twist by $24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.696.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.337153536.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.90699264.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.15501312.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.8226356490141696.17 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.625753767826996319210405216423877990678528.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.28403983392791185836308090622836736.1 | \(\Z/6\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | 1 | 1 | 3 | 1 | 1 | 1,1 | 3 | 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.