Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-7x+11\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-7xz^2+11z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-594x+6264\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1, 2\right) \) | $0.55519724858655603767738221254$ | $\infty$ |
| \( \left(2, 1\right) \) | $0.73493508504691552528073096628$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1:2:1]\) | $0.55519724858655603767738221254$ | $\infty$ |
| \([2:1:1]\) | $0.73493508504691552528073096628$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 54\right) \) | $0.55519724858655603767738221254$ | $\infty$ |
| \( \left(15, 27\right) \) | $0.73493508504691552528073096628$ | $\infty$ |
Integral points
\((-1,\pm 4)\), \((1,\pm 2)\), \((2,\pm 1)\), \((7,\pm 16)\), \((10,\pm 29)\), \((209,\pm 3014)\)
\([-1:\pm 4:1]\), \([1:\pm 2:1]\), \([2:\pm 1:1]\), \([7:\pm 16:1]\), \([10:\pm 29:1]\), \([209:\pm 3014:1]\)
\((-1,\pm 4)\), \((1,\pm 2)\), \((2,\pm 1)\), \((7,\pm 16)\), \((10,\pm 29)\), \((209,\pm 3014)\)
Invariants
| Conductor: | $N$ | = | \( 3328 \) | = | $2^{8} \cdot 13$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-6656$ | = | $-1 \cdot 2^{9} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( -\frac{85184}{13} \) | = | $-1 \cdot 2^{6} \cdot 11^{3} \cdot 13^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.53719688351048543229984425295$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0570572689304444143627683440$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.7338285195750366$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.1978628675282614$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.38232559247018185324382778116$ |
|
| Real period: | $\Omega$ | ≈ | $4.0712642719433539936160141203$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $3.1130970497468527730050324898 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.113097050 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.071264 \cdot 0.382326 \cdot 2}{1^2} \\ & \approx 3.113097050\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 8 | 9 | 0 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|---|
| $5$ | 5B | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1040 = 2^{4} \cdot 5 \cdot 13 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 391 & 790 \\ 0 & 807 \end{array}\right),\left(\begin{array}{rr} 1031 & 10 \\ 1030 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 985 & 921 \end{array}\right),\left(\begin{array}{rr} 911 & 10 \\ 395 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 787 & 10 \\ 325 & 93 \end{array}\right),\left(\begin{array}{rr} 561 & 10 \\ 725 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[1040])$ is a degree-$6440878080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1040\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 13 \) |
| $13$ | split multiplicative | $14$ | \( 256 = 2^{8} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 3328.c
consists of 2 curves linked by isogenies of
degree 5.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{16})^+\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.8624090508400001024.106 | \(\Z/4\Z\) | not in database |
| $12$ | 12.4.981348487528448.4 | \(\Z/10\Z\) | not in database |
| $20$ | 20.0.731633299112184496737658863616000000000000000.1 | \(\Z/5\Z\) | 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 | add | ord | ord | ord | ss | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | 2 | 2 | 2,4 | 3 | 2 | 2 | 2 | 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.