Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-7854701x+7460627299\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-7854701xz^2+7460627299z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10179692523x+348235722658278\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(558641, 417256980)$ | $4.5856996993943866106051356795$ | $\infty$ |
| $(-3193, 1596)$ | $0$ | $2$ |
Integral points
\( \left(-3193, 1596\right) \), \( \left(558641, 417256980\right) \), \( \left(558641, -417815622\right) \)
Invariants
| Conductor: | $N$ | = | \( 82810 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $6948161961870647336000$ | = | $2^{6} \cdot 5^{3} \cdot 7^{12} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{94376601570889}{12235496000} \) | = | $2^{-6} \cdot 5^{-3} \cdot 7^{-6} \cdot 11^{3} \cdot 13^{-1} \cdot 4139^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9195741090296580357648079076$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.66414435577123301518538781510$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9299956493074025$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.231533061815963$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.5856996993943866106051356795$ |
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| Real period: | $\Omega$ | ≈ | $0.12803754152884180888493377271$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2 \cdot 3 )\cdot1\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.091401176800148196544733202 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.091401177 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.128038 \cdot 4.585700 \cdot 96}{2^2} \\ & \approx 14.091401177\end{aligned}$$
Modular invariants
Modular form 82810.2.a.cn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6967296 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 4341 & 10912 \end{array}\right),\left(\begin{array}{rr} 1670 & 10917 \\ 4227 & 8 \end{array}\right),\left(\begin{array}{rr} 5919 & 7738 \\ 3206 & 5021 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right),\left(\begin{array}{rr} 3119 & 10908 \\ 7794 & 10847 \end{array}\right),\left(\begin{array}{rr} 5461 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3641 & 12 \\ 9100 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 41405 = 5 \cdot 7^{2} \cdot 13^{2} \) |
| $3$ | good | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 16562 = 2 \cdot 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 82810ca
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 910c3, its twist by $-91$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.203840.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{65}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.92840700771.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.175551900160000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.568788156518400.95 | \(\Z/12\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.10964985397162700412873689470112597284473000000000000.1 | \(\Z/18\Z\) | 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 | split | ord | nonsplit | add | ss | add | ss | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 1 | 1 | - | 1,3 | - | 3,1 | 5 | 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.