Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-507x+4381\)
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(homogenize, simplify) |
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\(y^2z=x^3-507xz^2+4381z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-507x+4381\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12, 5)$ | $1.1913135972585599372014835612$ | $\infty$ |
Integral points
\((12,\pm 5)\)
Invariants
| Conductor: | $N$ | = | \( 121680 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $49280400$ | = | $2^{4} \cdot 3^{6} \cdot 5^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{7311616}{25} \) | = | $2^{8} \cdot 5^{-2} \cdot 13^{4}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.33958868525733148279514508976$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.86825807884029458871713614278$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9881100332812158$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.5876460915240056$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1913135972585599372014835612$ |
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| Real period: | $\Omega$ | ≈ | $2.0153948847492451359485142503$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8019346600942480815570690804 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.801934660 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.015395 \cdot 1.191314 \cdot 2}{1^2} \\ & \approx 4.801934660\end{aligned}$$
Modular invariants
Modular form 121680.2.a.de
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46080 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 |
|---|---|---|
| $2$ | 2Cn | 2.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 211 & 264 \\ 216 & 271 \end{array}\right),\left(\begin{array}{rr} 647 & 516 \\ 642 & 509 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 772 & 775 \end{array}\right),\left(\begin{array}{rr} 301 & 3 \\ 561 & 4 \end{array}\right),\left(\begin{array}{rr} 259 & 0 \\ 0 & 779 \end{array}\right),\left(\begin{array}{rr} 777 & 4 \\ 776 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[780])$ is a degree-$4830658560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/780\Z)$.
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 | $2$ | \( 13520 = 2^{4} \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 24336 = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $38$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 121680bt consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6760a1, its twist by $12$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.169.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | add | add | split | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.