Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1177x-22786\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1177xz^2-22786z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1525419x-1040214042\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(41, -21)$ | $0$ | $2$ |
Integral points
\( \left(41, -21\right) \)
Invariants
| Conductor: | $N$ | = | \( 21021 \) | = | $3 \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-110380997727$ | = | $-1 \cdot 3^{8} \cdot 7^{6} \cdot 11 \cdot 13 $ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.82090353287462616450520186547$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15205154165303048804747450625$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8840474094399083$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.370489523442716$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.39649332660565027435842870442$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.79298665321130054871685740885 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.792986653 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.396493 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.792986653\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24576 |
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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 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48048 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1072 & 27461 \\ 11571 & 13714 \end{array}\right),\left(\begin{array}{rr} 34952 & 34321 \\ 29407 & 20602 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12881 & 20608 \\ 11354 & 19167 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 47950 & 48035 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 48044 & 48045 \end{array}\right),\left(\begin{array}{rr} 48033 & 16 \\ 48032 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 20608 \\ 22316 & 8709 \end{array}\right),\left(\begin{array}{rr} 32033 & 20608 \\ 2296 & 20721 \end{array}\right),\left(\begin{array}{rr} 20591 & 0 \\ 0 & 48047 \end{array}\right)$.
The torsion field $K:=\Q(E[48048])$ is a degree-$4284987369062400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48048\Z)$.
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 | $2$ | \( 7007 = 7^{2} \cdot 11 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7007 = 7^{2} \cdot 11 \cdot 13 \) |
| $7$ | additive | $26$ | \( 429 = 3 \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 21021.e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 429.b5, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ 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 |
| $2$ | \(\Q(\sqrt{1001}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-143})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{13})\) | \(\Z/8\Z\) | 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 |
| $8$ | 8.0.1004006004001.1 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 7 | 11 | 13 |
|---|---|---|---|---|---|
| Reduction type | ord | nonsplit | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 5 | 0 | - | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0 | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.