Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-1470828x+686569552\)
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(homogenize, simplify) |
\(y^2z=x^3-1470828xz^2+686569552z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-1470828x+686569552\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(716, 720)$ | $2.3672160487983303464515820400$ | $\infty$ |
$(698, 0)$ | $0$ | $2$ |
Integral points
\((186,\pm 20480)\), \( \left(698, 0\right) \), \((716,\pm 720)\)
Invariants
Conductor: | $N$ | = | \( 48960 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 17$ |
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Discriminant: | $\Delta$ | = | $6008889094963200$ | = | $2^{26} \cdot 3^{6} \cdot 5^{2} \cdot 17^{3} $ |
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j-invariant: | $j$ | = | \( \frac{1841373668746009}{31443200} \) | = | $2^{-8} \cdot 5^{-2} \cdot 17^{-3} \cdot 19^{3} \cdot 6451^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1571972478819511594909083990$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.56817033270797834966743759835$ |
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$abc$ quality: | $Q$ | ≈ | $0.9894109418057115$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.020726145010653$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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Mordell-Weil rank: | $r$ | = | $ 1$ |
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3672160487983303464515820400$ |
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Real period: | $\Omega$ | ≈ | $0.39024086142919344457025607786$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2\cdot1 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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Special value: | $ L'(E,1)$ | ≈ | $3.6951377202882882391497618896 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.695137720 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.390241 \cdot 2.367216 \cdot 16}{2^2} \\ & \approx 3.695137720\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 737280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 6 | 26 | 8 |
$3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
$17$ | $1$ | $I_{3}$ | nonsplit 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 | 4.6.0.3 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1019 & 2016 \\ 1008 & 1751 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 1940 & 2021 \end{array}\right),\left(\begin{array}{rr} 1529 & 2016 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 1816 & 21 \\ 675 & 1666 \end{array}\right),\left(\begin{array}{rr} 2017 & 24 \\ 2016 & 25 \end{array}\right),\left(\begin{array}{rr} 1359 & 2036 \\ 1340 & 1959 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 22 \\ 1394 & 1691 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$7219445760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $2$ | \( 153 = 3^{2} \cdot 17 \) |
$3$ | additive | $2$ | \( 320 = 2^{6} \cdot 5 \) |
$5$ | nonsplit multiplicative | $6$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
$17$ | nonsplit multiplicative | $18$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 48960.p
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170.a2, its twist by $24$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{6}) \) | \(\Z/6\Z\) | not in database |
$4$ | 4.0.39168.2 | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{6}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$6$ | 6.0.11520000.1 | \(\Z/6\Z\) | not in database |
$8$ | 8.4.5005166307840000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.443364212736.22 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.1534132224.12 | \(\Z/12\Z\) | not in database |
$12$ | 12.0.1194393600000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/8\Z\) | 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 |
$18$ | 18.6.9619848871348491895858790400000000.1 | \(\Z/18\Z\) | 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 |
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Reduction type | add | add | nonsplit | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | 1 | 1 | 3 | 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 | 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.