Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-631012065x-6098270939775\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-631012065xz^2-6098270939775z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-51111977292x-4445792851027824\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-14259, 0)$ | $0$ | $2$ |
| $(29005, 0)$ | $0$ | $2$ |
Integral points
\( \left(-14745, 0\right) \), \( \left(-14259, 0\right) \), \( \left(29005, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 87360 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $13539479709450240000000000$ | = | $2^{24} \cdot 3^{10} \cdot 5^{10} \cdot 7^{2} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{105997782562506306791694649}{51649016225625000000} \) | = | $2^{-6} \cdot 3^{-10} \cdot 5^{-10} \cdot 7^{-2} \cdot 13^{-4} \cdot 19^{3} \cdot 3361^{3} \cdot 7411^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7762927274766898212785348041$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7365719566367718571526866219$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0358851904498263$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.363458272329779$ | |||
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: | $\Omega$ | ≈ | $0.030123320307419240052227615089$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 320 $ = $ 2^{2}\cdot2\cdot( 2 \cdot 5 )\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.60246640614838480104455230178 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.602466406 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.030123 \cdot 1.000000 \cdot 320}{4^2} \\ & \approx 0.602466406\end{aligned}$$
Modular invariants
Modular form 87360.2.a.dl
For more coefficients, see the Downloads section to the right.
| Modular degree: | 44236800 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $5$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 4.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 4 \\ 482 & 9 \end{array}\right),\left(\begin{array}{rr} 837 & 4 \\ 836 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 209 & 836 \\ 418 & 831 \end{array}\right),\left(\begin{array}{rr} 281 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 4 \\ 674 & 9 \end{array}\right),\left(\begin{array}{rr} 417 & 838 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 29120 = 2^{6} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 5824 = 2^{6} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 12480 = 2^{6} \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 87360.dl
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730.b2, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{70}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-70})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7965941760000.44 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | nonsplit | split | split | nonsplit |
| $\lambda$-invariant(s) | - | 0 | 1 | 1 | 0 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.