Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+10025493x+3676666394\)
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(homogenize, simplify) |
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\(y^2z=x^3+10025493xz^2+3676666394z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+10025493x+3676666394\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4783, 401310)$ | $1.0676078379669814569472070010$ | $\infty$ |
| $(-362, 0)$ | $0$ | $2$ |
Integral points
\( \left(-362, 0\right) \), \((38,\pm 63700)\), \((1663,\pm 157950)\), \((4783,\pm 401310)\)
Invariants
| Conductor: | $N$ | = | \( 65520 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-70330436788824576000000$ | = | $-1 \cdot 2^{15} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{37321015309599759191}{23553520979625000} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{-3} \cdot 23^{3} \cdot 31^{3} \cdot 43^{3} \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0729826677946302955681166301$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8305293429006301404532618902$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0200279220273976$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.408018880600074$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0676078379669814569472070010$ |
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| Real period: | $\Omega$ | ≈ | $0.068038719795696948089067535177$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2299842788170202358690268269 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.229984279 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.068039 \cdot 1.067608 \cdot 288}{2^2} \\ & \approx 5.229984279\end{aligned}$$
Modular invariants
Modular form 65520.2.a.ca
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5971968 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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
$\left(\begin{array}{rr} 10 & 3 \\ 15873 & 32608 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9385 & 36 \\ 31794 & 31369 \end{array}\right),\left(\begin{array}{rr} 5062 & 3 \\ 10341 & 32452 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 32725 & 36 \\ 32724 & 37 \end{array}\right),\left(\begin{array}{rr} 16379 & 32724 \\ 32742 & 23011 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 19657 & 36 \\ 26226 & 649 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 20474 & 1329 \\ 1365 & 4094 \end{array}\right)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $2$ | \( 16 = 2^{4} \) |
| $5$ | split multiplicative | $6$ | \( 13104 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 65520.ca
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 2730.m6, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.1146600.5 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-26})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.14219703912960000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.151613669376.9 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.84140259840000.6 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.84140259840000.30 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.469935263779846154989143478322130611381993472.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.3357447872226838670189925399535112392704000000000000.1 | \(\Z/18\Z\) | 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 | add | split | nonsplit | ord | split | ord | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1 | 2 | 1 | 1 | 1,1 | 1,3 | 1 | 1 | 1 | 3 | 1,1 |
| $\mu$-invariant(s) | - | - | 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.