Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6237516x-6452700784\)
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(homogenize, simplify) |
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\(y^2z=x^3-6237516xz^2-6452700784z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6237516x-6452700784\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{17849979359744}{4902100225}, \frac{47452157581877227028}{343220547253375}\right) \) | $30.158038284323709353866726433$ | $\infty$ |
| \( \left(2908, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1249766304872476160:47452157581877227028:343220547253375]\) | $30.158038284323709353866726433$ | $\infty$ |
| \([2908:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{17849979359744}{4902100225}, \frac{47452157581877227028}{343220547253375}\right) \) | $30.158038284323709353866726433$ | $\infty$ |
| \( \left(2908, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(2908, 0\right) \)
\([2908:0:1]\)
\( \left(2908, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 52416 \) | = | $2^{6} \cdot 3^{2} \cdot 7 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2455777184397794869248$ | = | $-1 \cdot 2^{17} \cdot 3^{30} \cdot 7 \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{280880296871140514}{25701087819771} \) | = | $-1 \cdot 2 \cdot 3^{-24} \cdot 7^{-1} \cdot 13^{-1} \cdot 519793^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8471684613426327725060557402$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3159038112153220718006876163$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0089455216720034$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.401569163412892$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $30.158038284323709353866726433$ |
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| Real period: | $\Omega$ | ≈ | $0.047518148213529996539663751202$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.7322165320952236459228482775 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.732216532 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047518 \cdot 30.158038 \cdot 16}{2^2} \\ & \approx 5.732216532\end{aligned}$$
Modular invariants
Modular form 52416.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2359296 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 6 | 17 | 0 |
| $3$ | $4$ | $I_{24}^{*}$ | additive | -1 | 2 | 30 | 24 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.24.0.11 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 4270 & 4355 \end{array}\right),\left(\begin{array}{rr} 3821 & 4352 \\ 2730 & 3275 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 4364 & 4365 \end{array}\right),\left(\begin{array}{rr} 2512 & 5 \\ 579 & 4354 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4353 & 16 \\ 4352 & 17 \end{array}\right),\left(\begin{array}{rr} 3032 & 1 \\ 2095 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3260 & 3275 \\ 2025 & 2174 \end{array}\right),\left(\begin{array}{rr} 2911 & 4352 \\ 1448 & 4239 \end{array}\right)$.
The torsion field $K:=\Q(E[4368])$ is a degree-$324620255232$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4368\Z)$.
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 | $4$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $3$ | additive | $6$ | \( 5824 = 2^{6} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 52416eu
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2184j6, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-182}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-182})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-26})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{7})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.364024420171776.239 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | add | ord | nonsplit | ord | nonsplit | ord | ord | ss | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
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.