Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-983974635x-11953853994726\)
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(homogenize, simplify) |
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\(y^2z=x^3-983974635xz^2-11953853994726z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-983974635x-11953853994726\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(36246, 0)$ | $0$ | $2$ |
Integral points
\( \left(36246, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 134064 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-758303647990373661264248832$ | = | $-1 \cdot 2^{16} \cdot 3^{9} \cdot 7^{18} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{11108001800138902875}{79947274872976} \) | = | $-1 \cdot 2^{-4} \cdot 3^{6} \cdot 5^{3} \cdot 7^{-12} \cdot 19^{-2} \cdot 179^{3} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9900260548527289998619259623$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4999645832640447693455835414$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0300193263247077$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.246556148304249$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.013472149321100371926546867077$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.9399895022384535574227488591 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.939989502 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.013472 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 1.939989502\end{aligned}$$
Modular invariants
Modular form 134064.2.a.cp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 63700992 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $7$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1585 & 12 \\ 1584 & 13 \end{array}\right),\left(\begin{array}{rr} 748 & 217 \\ 567 & 944 \end{array}\right),\left(\begin{array}{rr} 237 & 1232 \\ 1414 & 559 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1546 & 1587 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1009 & 924 \\ 126 & 757 \end{array}\right),\left(\begin{array}{rr} 683 & 0 \\ 0 & 1595 \end{array}\right)$.
The torsion field $K:=\Q(E[1596])$ is a degree-$11914076160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1596\Z)$.
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 14896 = 2^{4} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $32$ | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 134064cy
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2394b2, its twist by $-84$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.477603.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.8342112022272.7 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2588134477824.57 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.228104625609.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.58394784155904.109 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 |
| $18$ | 18.0.208890268803780640999952843185158394157727744.2 | \(\Z/18\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|
| Reduction type | add | add | add | split |
| $\lambda$-invariant(s) | - | - | - | 1 |
| $\mu$-invariant(s) | - | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.