Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-192984987x-1123982734966\)
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(homogenize, simplify) |
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\(y^2z=x^3-192984987xz^2-1123982734966z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-192984987x-1123982734966\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(40573, 7605000)$ | $0$ | $4$ |
Integral points
\( \left(16198, 0\right) \), \((40573,\pm 7605000)\)
Invariants
| Conductor: | $N$ | = | \( 121680 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-85769379174375000000000000$ | = | $-1 \cdot 2^{12} \cdot 3^{7} \cdot 5^{16} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{55150149867714721}{5950927734375} \) | = | $-1 \cdot 3^{-1} \cdot 5^{-16} \cdot 13^{-1} \cdot 380641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7143450714194024101157953342$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1894170677946338869741968735$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0142537292135114$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.894445368123186$ | |||
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: | $\Omega$ | ≈ | $0.020130353436070167786419434030$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1024 $ = $ 2^{2}\cdot2^{2}\cdot2^{4}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.2883426199084907383308437779 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.288342620 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.020130 \cdot 1.000000 \cdot 1024}{4^2} \\ & \approx 1.288342620\end{aligned}$$
Modular invariants
Modular form 121680.2.a.ef
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33030144 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 16.48.0.34 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 2497 & 32 \\ 2512 & 513 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 3678 & 4235 \end{array}\right),\left(\begin{array}{rr} 6209 & 32 \\ 6208 & 33 \end{array}\right),\left(\begin{array}{rr} 4289 & 6208 \\ 5834 & 5727 \end{array}\right),\left(\begin{array}{rr} 2341 & 32 \\ 2356 & 513 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 6210 & 6209 \\ 2231 & 156 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4298 & 6237 \\ 3955 & 212 \end{array}\right)$.
The torsion field $K:=\Q(E[6240])$ is a degree-$309162147840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6240\Z)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 13520 = 2^{4} \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 24336 = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 121680.ef
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 195.a3, its twist by $156$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.15185664.1 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.900798402816.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.230604391120896.21 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.1639853447970816.63 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | add | add | split | add |
| $\lambda$-invariant(s) | - | - | 1 | - |
| $\mu$-invariant(s) | - | - | 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 $0$.