Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-6805617032x-216099982943424\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-6805617032xz^2-216099982943424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8820079674147x-10082228503013281314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-47664, 23832)$ | $0$ | $2$ |
Integral points
\( \left(-47664, 23832\right) \)
Invariants
| Conductor: | $N$ | = | \( 19110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $31579388527190999040000000$ | = | $2^{24} \cdot 3^{8} \cdot 5^{7} \cdot 7^{10} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{296304326013275547793071733369}{268420373544960000000} \) | = | $2^{-24} \cdot 3^{-8} \cdot 5^{-7} \cdot 7^{-4} \cdot 13^{-1} \cdot 59^{3} \cdot 112995371^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1911077896645285938304604127$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.2181527151368719412777840410$ |
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| $abc$ quality: | $Q$ | ≈ | $1.052714565604007$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.068260439433422$ | |||
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: | $\Omega$ | ≈ | $0.016621977940314292417566122057$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 112 $ = $ 2\cdot2\cdot7\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.46541538232880018769185141760 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.465415382 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.016622 \cdot 1.000000 \cdot 112}{2^2} \\ & \approx 0.465415382\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20643840 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1369 & 1368 \\ 3198 & 1375 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 848 & 3 \\ 2805 & 2 \end{array}\right),\left(\begin{array}{rr} 2188 & 1 \\ 751 & 6 \end{array}\right),\left(\begin{array}{rr} 3119 & 3632 \\ 1556 & 3607 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3634 & 3635 \end{array}\right),\left(\begin{array}{rr} 459 & 458 \\ 1378 & 3195 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $4$ | \( 3185 = 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3185 = 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 78 = 2 \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 19110p
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730k1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-91}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-35}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.79625.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.46356673636000000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.74170677817600.13 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1071483765625.4 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | add | split |
| $\lambda$-invariant(s) | 5 | 0 | 1 | - | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ 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 $0$.