Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-381916x-83523284\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-381916xz^2-83523284z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30935223x-60981279678\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-824775/2809, 266932292/148877)$ | $11.548662829521332562866370204$ | $\infty$ |
| $(-271, 0)$ | $0$ | $2$ |
Integral points
\( \left(-271, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 29040 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $542320423497388800$ | = | $2^{8} \cdot 3^{3} \cdot 5^{2} \cdot 11^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{13584145739344}{1195803675} \) | = | $2^{4} \cdot 3^{-3} \cdot 5^{-2} \cdot 11^{-6} \cdot 17^{3} \cdot 557^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1431956660911996663980673956$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48214990931871752142227419231$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9601937696230064$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.882286567246929$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.548662829521332562866370204$ |
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| Real period: | $\Omega$ | ≈ | $0.19312613854750781157002867828$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4606973153051809176627685310 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.460697315 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.193126 \cdot 11.548663 \cdot 8}{2^2} \\ & \approx 4.460697315\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 276480 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $4$ | $I_{6}^{*}$ | 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 | 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 \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 121 & 12 \\ 120 & 13 \end{array}\right),\left(\begin{array}{rr} 119 & 120 \\ 54 & 59 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 54 & 131 \\ 55 & 10 \end{array}\right),\left(\begin{array}{rr} 50 & 11 \\ 33 & 112 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 82 & 123 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$633600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
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 | $2$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 29040ce
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 660d2, its twist by $44$.
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 |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.5808.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.5749920000.5 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.4857532416.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1748711669760000.45 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.539725824.2 | \(\Z/12\Z\) | 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 |
| $16$ | 16.0.23595621172490797056.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.5636478432977148476745016616543846400000000.3 | \(\Z/18\Z\) | 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 | add | nonsplit | nonsplit | ord | add | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | 3 | 1 | 1 | - | 1 | 1,1 | 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 | 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.