Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-39951x-3708702\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-39951xz^2-3708702z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-51775875x-172877861250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{2181348}{7921}, \frac{1656999946}{704969}\right) \) | $11.822754970244421185992223202$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([194139972:1656999946:704969]\) | $11.822754970244421185992223202$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{78552291}{7921}, \frac{378955241964}{704969}\right) \) | $11.822754970244421185992223202$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 26450 \) | = | $2 \cdot 5^{2} \cdot 23^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1850448612500000$ | = | $-1 \cdot 2^{5} \cdot 5^{8} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{121945}{32} \) | = | $-1 \cdot 2^{-5} \cdot 5 \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6463664680377434999499307025$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.99433924821623159518728526889$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9433414068835279$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.298702258364273$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.822754970244421185992223202$ |
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| Real period: | $\Omega$ | ≈ | $0.16655853972963965233905071387$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9383616068505001836368706071 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.938361607 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.166559 \cdot 11.822755 \cdot 2}{1^2} \\ & \approx 3.938361607\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 118800 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $23$ | $2$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $5$ | 5B.4.1 | 5.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 552 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 1680 \\ 1080 & 1081 \end{array}\right),\left(\begin{array}{rr} 1726 & 345 \\ 1725 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2070 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1380 & 1 \end{array}\right),\left(\begin{array}{rr} 2071 & 690 \\ 2415 & 691 \end{array}\right),\left(\begin{array}{rr} 1559 & 0 \\ 0 & 2759 \end{array}\right),\left(\begin{array}{rr} 921 & 2530 \\ 1840 & 921 \end{array}\right),\left(\begin{array}{rr} 1841 & 920 \\ 1840 & 921 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1680 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1242 \\ 690 & 1381 \end{array}\right),\left(\begin{array}{rr} 1381 & 690 \\ 1725 & 691 \end{array}\right),\left(\begin{array}{rr} 1839 & 1150 \\ 920 & 1103 \end{array}\right),\left(\begin{array}{rr} 1 & 276 \\ 1380 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$24622202880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 13225 = 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $14$ | \( 529 = 23^{2} \) |
| $23$ | additive | $266$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 26450.i
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50.b3, its twist by $-115$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/3\Z\) | 2.0.23.1-2500.2-a2 |
| $2$ | \(\Q(\sqrt{-115}) \) | \(\Z/5\Z\) | 2.0.115.1-100.1-b2 |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-23})\) | \(\Z/15\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.3285090000.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.486680000.3 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.2433400000.4 | \(\Z/10\Z\) | 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 |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/30\Z\) | not in database |
| $18$ | 18.0.43612715304227932212937500000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.580847007098080935936000000000000.2 | \(\Z/6\Z\) | not in database |
| $20$ | 20.4.192907225404162891209125518798828125.1 | \(\Z/5\Z\) | 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 | nonsplit | ord | add | ord | ord | ord | ord | ord | add | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 3 | - | 1 | 5 | 1 | 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 |
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.