Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-373x+2623\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-373xz^2+2623z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-484083x+129636558\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(13, 7\right) \) | $0.53264658095574456192746745541$ | $\infty$ |
| \( \left(\frac{43}{4}, -\frac{43}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([13:7:1]\) | $0.53264658095574456192746745541$ | $\infty$ |
| \([86:-43:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(483, 2916\right) \) | $0.53264658095574456192746745541$ | $\infty$ |
| \( \left(402, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(11, -4\right) \), \( \left(11, -7\right) \), \( \left(13, 7\right) \), \( \left(13, -20\right) \), \( \left(31, 133\right) \), \( \left(31, -164\right) \)
\([11:-4:1]\), \([11:-7:1]\), \([13:7:1]\), \([13:-20:1]\), \([31:133:1]\), \([31:-164:1]\)
\((411,\pm 324)\), \((483,\pm 2916)\), \((1131,\pm 32076)\)
Invariants
| Conductor: | $N$ | = | \( 210 \) | = | $2 \cdot 3 \cdot 5 \cdot 7$ |
|
| Minimal Discriminant: | $\Delta$ | = | $5670$ | = | $2 \cdot 3^{4} \cdot 5 \cdot 7 $ |
|
| j-invariant: | $j$ | = | \( \frac{5763259856089}{5670} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{-1} \cdot 7^{-1} \cdot 17929^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.013583148081496443568986081444$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.013583148081496443568986081444$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9849151554257001$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.495031511708493$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.53264658095574456192746745541$ |
|
| Real period: | $\Omega$ | ≈ | $3.5846020263461843307559188552$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $0.95466300671016443659063754511 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 0.954663007 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.584602 \cdot 0.532647 \cdot 2}{2^2} \\ & \approx 0.954663007\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | 8.12.0.8 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 176 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 528 & 113 \\ 307 & 294 \end{array}\right),\left(\begin{array}{rr} 736 & 323 \\ 739 & 768 \end{array}\right),\left(\begin{array}{rr} 248 & 3 \\ 365 & 2 \end{array}\right),\left(\begin{array}{rr} 281 & 8 \\ 284 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 834 & 835 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 35 = 5 \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 210d
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{70}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{35}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7710244864000000.17 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.609892416000000.8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.416179814400.17 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.4253299470000.4 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | nonsplit | nonsplit | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | 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.