Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-1168545x+139637025\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-1168545xz^2+139637025z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-94652172x+101511434736\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2425, 107520)$ | $2.4587640088777866941619187654$ | $\infty$ |
| $(121, 0)$ | $0$ | $2$ |
Integral points
\( \left(121, 0\right) \), \((2425,\pm 107520)\)
Invariants
| Conductor: | $N$ | = | \( 87360 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $93744955442331648000$ | = | $2^{24} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{673163386034885929}{357608625192000} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{-3} \cdot 13^{-6} \cdot 29^{3} \cdot 47^{3} \cdot 643^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5236765234605754498704940116$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4839557526206574857446458294$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0044067734582287$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.704551638550009$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4587640088777866941619187654$ |
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| Real period: | $\Omega$ | ≈ | $0.16666014625165844525824927003$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 2^{2}\cdot1\cdot3\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.3760034477219471780661212128 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.376003448 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.166660 \cdot 2.458764 \cdot 72}{2^2} \\ & \approx 7.376003448\end{aligned}$$
Modular invariants
Modular form 87360.2.a.ds
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2985984 |
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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 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 16389 \\ 16227 & 28432 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 32725 & 36 \\ 32724 & 37 \end{array}\right),\left(\begin{array}{rr} 16379 & 32724 \\ 32742 & 32111 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 12313 & 15018 \\ 17082 & 19819 \end{array}\right),\left(\begin{array}{rr} 18742 & 3 \\ 2421 & 32452 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 25701 & 32608 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 17665 & 36 \\ 4074 & 31369 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$175294937825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 | $2$ | \( 105 = 3 \cdot 5 \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 64 = 2^{6} \) |
| $5$ | split multiplicative | $6$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | 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, 3 and 6.
Its isogeny class 87360.ds
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 2730.m3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.4542720.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-70})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.497871360000.14 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.20636304998400.7 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.185726744985600.1 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.8911365002047453013127461514849291593614098432.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.5157039931740424197411725413685932635193344000000000000.1 | \(\Z/18\Z\) | 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 | nonsplit | split | split | ord | nonsplit | ord | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 1 | 0 | 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.