Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-119706713163x-15941348561057081\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-119706713163xz^2-15941348561057081z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1915307410611x-1020248223215063794\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(702289549318041324700071/17653590528244225, 588468733919332734360572172975774599/2345576346801977717893375)$ | $49.075911009118697537608545302$ | $\infty$ |
| $(-799021/4, 799021/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 126126 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $25164088126425059353542$ | = | $2 \cdot 3^{10} \cdot 7^{14} \cdot 11 \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{2211889682389423686563629156897}{293403593785302} \) | = | $2^{-1} \cdot 3^{-4} \cdot 7^{-8} \cdot 11^{-1} \cdot 13^{-4} \cdot 13029302113^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.6264779835238937457424392426$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.1042167646621822474921402524$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0439991800655393$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.504321781242278$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $49.075911009118697537608545302$ |
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| Real period: | $\Omega$ | ≈ | $0.0081165212138202562817039831150$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.1866053823445357542760471669 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.186605382 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.008117 \cdot 49.075911 \cdot 32}{2^2} \\ & \approx 3.186605382\end{aligned}$$
Modular invariants
Modular form 126126.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 301989888 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $7$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{4}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3550 & 309 \\ 2697 & 1858 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 3692 & 3693 \end{array}\right),\left(\begin{array}{rr} 3681 & 16 \\ 3680 & 17 \end{array}\right),\left(\begin{array}{rr} 1231 & 0 \\ 0 & 3695 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2111 & 2448 \\ 3336 & 1103 \end{array}\right),\left(\begin{array}{rr} 2788 & 309 \\ 2007 & 1858 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3598 & 3683 \end{array}\right),\left(\begin{array}{rr} 3376 & 2469 \\ 2979 & 3682 \end{array}\right)$.
The torsion field $K:=\Q(E[3696])$ is a degree-$163499212800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3696\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 14014 = 2 \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $7$ | additive | $32$ | \( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 11466 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 126126bz
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 6006bd5, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-462}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-21}, \sqrt{22})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-21})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{-21})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.186606965293056.81 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | add | ord | add | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | - | 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.