Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-6942x+316458\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-6942xz^2+316458z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8996859x+14899625622\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(316/9, 8770/27)$ | $6.6758240425430423452340314657$ | $\infty$ |
| $(-101, 50)$ | $0$ | $2$ |
Integral points
\( \left(-101, 50\right) \)
Invariants
| Conductor: | $N$ | = | \( 123981 \) | = | $3 \cdot 11 \cdot 13 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-22646422399887$ | = | $-1 \cdot 3^{8} \cdot 11 \cdot 13 \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1532808577}{938223} \) | = | $-1 \cdot 3^{-8} \cdot 11^{-1} \cdot 13^{-1} \cdot 1153^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2645551303750775520772928027$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15205154165303048804747450624$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8840474094399083$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3144288480677084$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.6758240425430423452340314657$ |
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| Real period: | $\Omega$ | ≈ | $0.62674506627966357474881957448$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.1840397820150106986362008264 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.184039782 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.626745 \cdot 6.675824 \cdot 4}{2^2} \\ & \approx 4.184039782\end{aligned}$$
Modular invariants
Modular form 123981.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 294912 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 116688 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1072 & 96101 \\ 59619 & 61762 \end{array}\right),\left(\begin{array}{rr} 1 & 27472 \\ 42908 & 15573 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 116673 & 16 \\ 116672 & 17 \end{array}\right),\left(\begin{array}{rr} 38897 & 27472 \\ 91528 & 103089 \end{array}\right),\left(\begin{array}{rr} 61775 & 0 \\ 0 & 116687 \end{array}\right),\left(\begin{array}{rr} 62408 & 89233 \\ 111775 & 102970 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 116590 & 116675 \end{array}\right),\left(\begin{array}{rr} 60929 & 27472 \\ 66266 & 60351 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 116684 & 116685 \end{array}\right)$.
The torsion field $K:=\Q(E[116688])$ is a degree-$166502366340710400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/116688\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 11271 = 3 \cdot 13 \cdot 17^{2} \) |
| $13$ | split multiplicative | $14$ | \( 9537 = 3 \cdot 11 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 429 = 3 \cdot 11 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 123981h
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 429b1, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-143}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2431}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-143})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{17})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{17})\) | \(\Z/8\Z\) | 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 |
| $8$ | 8.0.34925275077121.3 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | nonsplit | ord | ss | split | split | add | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | 3 | 1 | 1,1 | 2 | 2 | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.