Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-32114x+2211536\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-32114xz^2+2211536z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-41619123x+103306292622\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(78, 388)$ | $1.4168511070097773871190038307$ | $\infty$ |
| $(-207, 103)$ | $0$ | $2$ |
| $(105, -53)$ | $0$ | $2$ |
Integral points
\( \left(-207, 103\right) \), \( \left(78, 388\right) \), \( \left(78, -467\right) \), \( \left(105, -53\right) \), \( \left(131, 441\right) \), \( \left(131, -573\right) \), \( \left(144, 688\right) \), \( \left(144, -833\right) \), \( \left(9231, 882127\right) \), \( \left(9231, -891359\right) \)
Invariants
| Conductor: | $N$ | = | \( 96330 \) | = | $2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $1568230244100$ | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{6} \cdot 19^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{758800078561}{324900} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 19^{-2} \cdot 1303^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3004163962578024944187943094$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.017941717527034126392050588618$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9387086030721215$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7248546617779246$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4168511070097773871190038307$ |
|
| Real period: | $\Omega$ | ≈ | $0.83232689255816386779183742833$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.7171331164601700808254607305 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.717133116 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.832327 \cdot 1.416851 \cdot 64}{4^2} \\ & \approx 4.717133116\end{aligned}$$
Modular invariants
Modular form 96330.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 442368 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 29640 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 10323 & 22802 \\ 5278 & 6839 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9881 & 15964 \\ 12922 & 2289 \end{array}\right),\left(\begin{array}{rr} 28731 & 22802 \\ 12766 & 6839 \end{array}\right),\left(\begin{array}{rr} 11399 & 0 \\ 0 & 29639 \end{array}\right),\left(\begin{array}{rr} 14821 & 15964 \\ 22802 & 2289 \end{array}\right),\left(\begin{array}{rr} 573 & 15964 \\ 2288 & 25091 \end{array}\right),\left(\begin{array}{rr} 29637 & 4 \\ 29636 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[29640])$ is a degree-$2379002727628800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29640\Z)$.
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 | $4$ | \( 169 = 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 32110 = 2 \cdot 5 \cdot 13^{2} \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 19266 = 2 \cdot 3 \cdot 13^{2} \cdot 19 \) |
| $13$ | additive | $86$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 96330bf
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 570m2, its twist by $13$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-78}, \sqrt{570})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-13}, \sqrt{-95})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | nonsplit | ord | ord | add | ord | split | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 2 | 1 | 1 | 1 | - | 1 | 2 | 1 | 1 | 1 | 3 | 1 | 1 | 1,1 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.