Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-225505x-144561025\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-225505xz^2-144561025z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18265932x-105330189456\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1085, 29820)$ | $2.9031123048535447813103516217$ | $\infty$ |
| $(1715, 67200)$ | $0$ | $4$ |
Integral points
\( \left(665, 0\right) \), \((1085,\pm 29820)\), \((1715,\pm 67200)\), \((3290,\pm 186375)\), \((20915,\pm 3024000)\)
Invariants
| Conductor: | $N$ | = | \( 87360 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-8284579430400000000$ | = | $-1 \cdot 2^{23} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{4837870546133689}{31603162500000} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-4} \cdot 5^{-8} \cdot 7^{-4} \cdot 13^{-1} \cdot 169129^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3135218923477418020734278237$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2738011215078238379475796415$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9837367724122568$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.491251584672357$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9031123048535447813103516217$ |
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| Real period: | $\Omega$ | ≈ | $0.097560337177425928881446790182$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 512 $ = $ 2^{2}\cdot2^{2}\cdot2^{3}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.0633156904142707790408740918 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.063315690 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.097560 \cdot 2.903112 \cdot 512}{4^2} \\ & \approx 9.063315690\end{aligned}$$
Modular invariants
Modular form 87360.2.a.gt
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1966080 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{13}^{*}$ | additive | 1 | 6 | 23 | 5 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1457 & 8 \\ 1460 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 816 & 1903 \\ 1373 & 1386 \end{array}\right),\left(\begin{array}{rr} 1249 & 8 \\ 628 & 33 \end{array}\right),\left(\begin{array}{rr} 1912 & 827 \\ 1915 & 1944 \end{array}\right),\left(\begin{array}{rr} 848 & 3 \\ 1349 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $4$ | \( 13 \) |
| $3$ | split multiplicative | $4$ | \( 29120 = 2^{6} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 12480 = 2^{6} \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split 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 and 4.
Its isogeny class 87360dr
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730d4, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.11741184.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.80980417183744.23 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | split | split | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 2 | 2 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 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.