Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-33836x+2394576\)
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(homogenize, simplify) |
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\(y^2z=x^3-33836xz^2+2394576z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-33836x+2394576\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-54, 2016\right) \) | $2.0629346706849873936683941524$ | $\infty$ |
| \( \left(74, 544\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-54:2016:1]\) | $2.0629346706849873936683941524$ | $\infty$ |
| \([74:544:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-54, 2016\right) \) | $2.0629346706849873936683941524$ | $\infty$ |
| \( \left(74, 544\right) \) | $0$ | $4$ |
Integral points
\((-54,\pm 2016)\), \((74,\pm 544)\), \( \left(108, 0\right) \), \((380,\pm 6664)\)
\([-54:\pm 2016:1]\), \([74:\pm 544:1]\), \([108:0:1]\), \([380:\pm 6664:1]\)
\((-54,\pm 2016)\), \((74,\pm 544)\), \( \left(108, 0\right) \), \((380,\pm 6664)\)
Invariants
| Conductor: | $N$ | = | \( 7616 \) | = | $2^{6} \cdot 7 \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $2145663844352$ | = | $2^{19} \cdot 7^{2} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{16342588257633}{8185058} \) | = | $2^{-1} \cdot 3^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 17^{-4} \cdot 769^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3188498824964950401489310969$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.27912911165657707602308291471$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1194511520331367$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.799889603564516$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0629346706849873936683941524$ |
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| Real period: | $\Omega$ | ≈ | $0.81274383326212463269196277930$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3532748640437107654150962222 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.353274864 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.812744 \cdot 2.062935 \cdot 32}{4^2} \\ & \approx 3.353274864\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12288 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $4$ | $I_{4}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.24.0.49 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16 & 77 \\ 47 & 18 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 62 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 129 & 8 \\ 128 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 130 & 131 \end{array}\right),\left(\begin{array}{rr} 105 & 8 \\ 12 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$2506752$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 | $4$ | \( 1 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1088 = 2^{6} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 448 = 2^{6} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 7616i
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 238c3, its twist by $-8$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-4 + \sqrt{-34}})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.40282095616.10 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1401249857536.7 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.821386940416.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | ss | ord | nonsplit | ss | ord | split | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1,1 | 1 | 1 | 1,1 | 1 | 2 | 3 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 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.