Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-3068x+70980\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-3068xz^2+70980z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-248535x+52489998\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(52, 234)$ | $0$ | $6$ |
Integral points
\( \left(-65, 0\right) \), \((16,\pm 162)\), \((52,\pm 234)\)
Invariants
| Conductor: | $N$ | = | \( 2652 \) | = | $2^{2} \cdot 3 \cdot 13 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-390868478208$ | = | $-1 \cdot 2^{8} \cdot 3^{12} \cdot 13^{2} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( -\frac{12479332642000}{1526829993} \) | = | $-1 \cdot 2^{4} \cdot 3^{-12} \cdot 5^{3} \cdot 7^{3} \cdot 13^{-2} \cdot 17^{-1} \cdot 263^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.95914583092766312088149296934$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.49704771055436624793667155503$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9159504743302306$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.553057047239804$ | |||
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: | $\Omega$ | ≈ | $0.92203858240096916174986260534$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 3\cdot( 2^{2} \cdot 3 )\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.8440771648019383234997252107 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.844077165 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.922039 \cdot 1.000000 \cdot 72}{6^2} \\ & \approx 1.844077165\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2652 = 2^{2} \cdot 3 \cdot 13 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 613 & 12 \\ 1026 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 222 & 1117 \\ 221 & 1106 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 893 & 2 \\ 942 & 13 \end{array}\right),\left(\begin{array}{rr} 2641 & 12 \\ 2640 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2602 & 2643 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 166 & 3 \\ 2469 & 2644 \end{array}\right)$.
The torsion field $K:=\Q(E[2652])$ is a degree-$98545434624$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2652\Z)$.
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 | $2$ | \( 17 \) |
| $3$ | split multiplicative | $4$ | \( 884 = 2^{2} \cdot 13 \cdot 17 \) |
| $13$ | split multiplicative | $14$ | \( 204 = 2^{2} \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 2652f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.2.45968.3 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.1030511497392.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.9770775678976.16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.1069350692061184.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.2190624242742007488.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.27908746691209002982923109495192584726320775168.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 13 | 17 |
|---|---|---|---|---|
| Reduction type | add | split | split | nonsplit |
| $\lambda$-invariant(s) | - | 1 | 1 | 0 |
| $\mu$-invariant(s) | - | 0 | 0 | 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 $0$.