Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-64783x+6358312\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-64783xz^2+6358312z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-5247450x+4619467125\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(152, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([152:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1365, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(152, 0\right) \)
\([152:0:1]\)
\( \left(152, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 101400 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 13^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $54301601250000$ | = | $2^{4} \cdot 3^{2} \cdot 5^{7} \cdot 13^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{24918016}{45} \) | = | $2^{11} \cdot 3^{-2} \cdot 5^{-1} \cdot 23^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5274717291778151825899912729$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.79077096595665180920954282165$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9923662209495213$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.890924071685363$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.62984554331880333124064943246$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.5193821732752133249625977299 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.519382173 \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 0.629846 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.519382173\end{aligned}$$
Modular invariants
Modular form 101400.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 368640 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|---|
| $2$ | 2B | 16.24.0.6 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 1456 \\ 2340 & 2341 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3022 & 3107 \end{array}\right),\left(\begin{array}{rr} 1613 & 1456 \\ 2704 & 885 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 2768 & 715 \\ 2925 & 974 \end{array}\right),\left(\begin{array}{rr} 2445 & 1456 \\ 1274 & 2939 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 3116 & 3117 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 33800 = 2^{3} \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 101400cb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 120a1, its twist by $65$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{39})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{65})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.148060224000000.74 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1827904000000.24 | \(\Z/8\Z\) | not in database |
| $8$ | 8.8.370150560000.1 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 5 | 13 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | add | add |
| $\lambda$-invariant(s) | - | 0 | - | - |
| $\mu$-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 $0$.