Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-388522x+92339127\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-388522xz^2+92339127z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-503524539x+4315727185686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(209, 4403)$ | $0.11656550102421204768403524294$ | $\infty$ |
| $(10897/9, 993433/27)$ | $1.7970550839937048267000352909$ | $\infty$ |
Integral points
\( \left(-711, 3483\right) \), \( \left(-711, -2773\right) \), \( \left(-343, 13787\right) \), \( \left(-343, -13445\right) \), \( \left(-183, 12635\right) \), \( \left(-183, -12453\right) \), \( \left(209, 4403\right) \), \( \left(209, -4613\right) \), \( \left(255, 3023\right) \), \( \left(255, -3279\right) \), \( \left(329, 347\right) \), \( \left(329, -677\right) \), \( \left(393, 539\right) \), \( \left(393, -933\right) \), \( \left(405, 1071\right) \), \( \left(405, -1477\right) \), \( \left(497, 4475\right) \), \( \left(497, -4973\right) \), \( \left(503, 4697\right) \), \( \left(503, -5201\right) \), \( \left(1865, 75611\right) \), \( \left(1865, -77477\right) \), \( \left(3123, 169757\right) \), \( \left(3123, -172881\right) \), \( \left(27257, 4485355\right) \), \( \left(27257, -4512613\right) \)
Invariants
| Conductor: | $N$ | = | \( 42826 \) | = | $2 \cdot 7^{2} \cdot 19 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $57036813950255104$ | = | $2^{21} \cdot 7^{6} \cdot 19 \cdot 23^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{55129288688387857}{484804919296} \) | = | $2^{-21} \cdot 19^{-1} \cdot 23^{-3} \cdot 43^{3} \cdot 53^{3} \cdot 167^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0388542854271889228235142095$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0658992108995322702708378378$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9640066419360215$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.709272027526846$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.20833056491910594337806743799$ |
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| Real period: | $\Omega$ | ≈ | $0.35428892674457910689942730898$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 126 $ = $ ( 3 \cdot 7 )\cdot2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $9.2999607439135203483961270874 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.299960744 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.354289 \cdot 0.208331 \cdot 126}{1^2} \\ & \approx 9.299960744\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 483840 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 73416 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 55063 & 41958 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 73411 & 6 \\ 73410 & 7 \end{array}\right),\left(\begin{array}{rr} 21414 & 20545 \\ 33649 & 67299 \end{array}\right),\left(\begin{array}{rr} 36709 & 41958 \\ 57687 & 52459 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 62927 & 0 \\ 0 & 73415 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 41497 & 41958 \\ 72051 & 52459 \end{array}\right),\left(\begin{array}{rr} 42505 & 41958 \\ 1659 & 52459 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[73416])$ is a degree-$305573750353428480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/73416\Z)$.
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 | $4$ | \( 21413 = 7^{2} \cdot 19 \cdot 23 \) |
| $3$ | good | $2$ | \( 931 = 7^{2} \cdot 19 \) |
| $7$ | additive | $26$ | \( 437 = 19 \cdot 23 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 1862 = 2 \cdot 7^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 42826n
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 874f1, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.3496.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.42728167936.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1206902781.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.4192151488.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.44846925612516485805095048707089072058368.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.6.24628105425206850134701226387889759869730816.1 | \(\Z/6\Z\) | 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 | ord | add | ord | ord | ss | nonsplit | split | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | 2 | 2 | - | 4 | 2 | 2,2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.