Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-7394371x-8411682856\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-7394371xz^2-8411682856z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9583104195x-392426726005314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9454, 870878)$ | $3.8704971323334669714515948993$ | $\infty$ |
| $(16065883/324, 64152143875/5832)$ | $14.677921408315496940908660391$ | $\infty$ |
| $(12679/4, -12683/8)$ | $0$ | $2$ |
Integral points
\( \left(9454, 870878\right) \), \( \left(9454, -880333\right) \)
Invariants
| Conductor: | $N$ | = | \( 106134 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-4687097164437865958802$ | = | $-1 \cdot 2 \cdot 3^{2} \cdot 7^{6} \cdot 19^{12} $ |
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| j-invariant: | $j$ | = | \( -\frac{8078253774625}{846825858} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 5^{3} \cdot 19^{-6} \cdot 4013^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8972502003409522883911337177$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.45207563623007540583394363003$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0101536552434738$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.11808080977086$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $56.262357035023774931845924461$ |
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| Real period: | $\Omega$ | ≈ | $0.045506953134743290994996774845$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.241313779372085702962812581 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.241313779 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.045507 \cdot 56.262357 \cdot 16}{2^2} \\ & \approx 10.241313779\end{aligned}$$
Modular invariants
Modular form 106134.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7464960 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 | 8.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2183 & 2268 \\ 3066 & 839 \end{array}\right),\left(\begin{array}{rr} 2129 & 924 \\ 994 & 2353 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2290 & 1827 \\ 1113 & 904 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3181 & 12 \\ 3180 & 13 \end{array}\right),\left(\begin{array}{rr} 666 & 1057 \\ 2261 & 1730 \end{array}\right),\left(\begin{array}{rr} 2279 & 0 \\ 0 & 3191 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3142 & 3183 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$190625218560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $4$ | \( 17689 = 7^{2} \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 35378 = 2 \cdot 7^{2} \cdot 19^{2} \) |
| $7$ | additive | $26$ | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 106134.z
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 114.c2, its twist by $133$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-399}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.5094432.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-399})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.2222733795408.5 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1661007193767936.116 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.25953237402624.21 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.96869093591377204460811555805816666097630976.1 | \(\Z/18\Z\) | 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 | split | ss | add | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | 3 | 2,2 | - | 2,2 | 2 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 1 | 1 | 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.