Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+352354920x-2007529902000\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+352354920xz^2-2007529902000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+456651976293x-93664685063640906\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7350, 985950)$ | $8.1394609687263747860865505011$ | $\infty$ |
| $(26160, 4998180)$ | $0$ | $4$ |
Integral points
\( \left(7350, 985950\right) \), \( \left(7350, -993300\right) \), \( \left(26160, 4998180\right) \), \( \left(26160, -5024340\right) \)
Invariants
| Conductor: | $N$ | = | \( 82110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-4540848316592979232425603600$ | = | $-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17^{16} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4837987390362436347081585367679}{4540848316592979232425603600} \) | = | $2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 17^{-16} \cdot 23^{-2} \cdot 71039^{3} \cdot 238081^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9941002091891954044324058131$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.9941002091891954044324058131$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0168811917145575$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.243832285107038$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.1394609687263747860865505011$ |
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| Real period: | $\Omega$ | ≈ | $0.023806154905303724018461847073$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1024 $ = $ 2^{2}\cdot2\cdot2\cdot2\cdot2^{4}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.401233194699109664652645957 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.401233195 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.023806 \cdot 8.139461 \cdot 1024}{4^2} \\ & \approx 12.401233195\end{aligned}$$
Modular invariants
Modular form 82110.2.a.bs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 58720256 |
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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 semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $23$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 16.96.0.25 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1313760 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 25 & 16 \\ 1312024 & 1312649 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1199549 & 24 \\ 631806 & 2885 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 563069 & 8 \\ 374334 & 1313477 \end{array}\right),\left(\begin{array}{rr} 437945 & 32 \\ 441302 & 4329 \end{array}\right),\left(\begin{array}{rr} 1236481 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25 & 32 \\ 266134 & 4329 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 1312374 & 1148653 \end{array}\right),\left(\begin{array}{rr} 1313729 & 32 \\ 1313728 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 942879 & 81223 \end{array}\right)$.
The torsion field $K:=\Q(E[1313760])$ is a degree-$497724242914852208640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1313760\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \) |
| $5$ | split multiplicative | $6$ | \( 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 \) |
| $17$ | split multiplicative | $18$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 82110.bs
consists of 8 curves linked by isogenies of
degrees dividing 16.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{4830})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{322})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | 4.2.28566000.1 | \(\Z/16\Z\) | not in database |
| $4$ | 4.0.10584000.3 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.816016356000000.57 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.0.1792336896000000.154 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | split | split | split | nonsplit | ord | ord | split | ord | nonsplit | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 10 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.