Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5196x+156240\)
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(homogenize, simplify) |
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\(y^2z=x^3-5196xz^2+156240z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5196x+156240\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 121\right) \) | $0.71969231553826413146810909566$ | $\infty$ |
| \( \left(-84, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37:121:1]\) | $0.71969231553826413146810909566$ | $\infty$ |
| \([-84:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 121\right) \) | $0.71969231553826413146810909566$ | $\infty$ |
| \( \left(-84, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-84, 0\right) \), \((37,\pm 121)\), \((48,\pm 132)\), \((444,\pm 9240)\)
\([-84:0:1]\), \([37:\pm 121:1]\), \([48:\pm 132:1]\), \([444:\pm 9240:1]\)
\( \left(-84, 0\right) \), \((37,\pm 121)\), \((48,\pm 132)\), \((444,\pm 9240)\)
Invariants
| Conductor: | $N$ | = | \( 6336 \) | = | $2^{6} \cdot 3^{2} \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1567363792896$ | = | $-1 \cdot 2^{15} \cdot 3^{3} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{17535471192}{1771561} \) | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 11^{-6} \cdot 433^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0800417646972045951843562995$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.061045283169754464435995161553$ |
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| $abc$ quality: | $Q$ | ≈ | $1.065125198948697$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.277065355490354$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.71969231553826413146810909566$ |
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| Real period: | $\Omega$ | ≈ | $0.82498127559906303588565772289$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5623961070696031074915569410 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.562396107 \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 0.824981 \cdot 0.719692 \cdot 24}{2^2} \\ & \approx 3.562396107\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3Nn | 3.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $72$, genus $3$, and generators
$\left(\begin{array}{rr} 35 & 194 \\ 188 & 37 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 105 & 256 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 145 & 12 \\ 78 & 73 \end{array}\right),\left(\begin{array}{rr} 253 & 12 \\ 252 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 8 \\ 220 & 225 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 184 & 7 \\ 65 & 244 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$13516800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 3 \) |
| $3$ | additive | $6$ | \( 64 = 2^{6} \) |
| $11$ | split multiplicative | $12$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6336.s
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3168.c2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.104544.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.369975361536.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.699484667904.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.573308928.1 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.328683126924509184.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.5258930030792146944.3 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ss | split | ss | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | 1,1 | 2 | 1,1 | 1 | 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 |
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.