Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-484072x+155404144\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-484072xz^2+155404144z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-39209859x+113171991426\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-732, 10816\right) \) | $1.2380661412650862487856608293$ | $\infty$ |
| \( \left(282, 6422\right) \) | $2.3166253772474057010128401108$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-732:10816:1]\) | $1.2380661412650862487856608293$ | $\infty$ |
| \([282:6422:1]\) | $2.3166253772474057010128401108$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-6591, 292032\right) \) | $1.2380661412650862487856608293$ | $\infty$ |
| \( \left(2535, 173394\right) \) | $2.3166253772474057010128401108$ | $\infty$ |
Integral points
\((-732,\pm 10816)\), \((282,\pm 6422)\), \((450,\pm 5338)\), \((740,\pm 14208)\), \((4169,\pm 265668)\)
\([-732:\pm 10816:1]\), \([282:\pm 6422:1]\), \([450:\pm 5338:1]\), \([740:\pm 14208:1]\), \([4169:\pm 265668:1]\)
\((-732,\pm 10816)\), \((282,\pm 6422)\), \((450,\pm 5338)\), \((740,\pm 14208)\), \((4169,\pm 265668)\)
Invariants
| Conductor: | $N$ | = | \( 137904 \) | = | $2^{4} \cdot 3 \cdot 13^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $-3151771755300519936$ | = | $-1 \cdot 2^{18} \cdot 3 \cdot 13^{8} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{3754462153}{943296} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-1} \cdot 13 \cdot 17^{-3} \cdot 661^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2673007225363739214124806943$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13581269633126254537374305487$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9003384877332229$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.330132798079682$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7181197834219738173829696145$ |
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| Real period: | $\Omega$ | ≈ | $0.24029028431339790734728907243$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2^{2}\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.8376533066760515895587731514 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.837653307 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \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 0.240290 \cdot 2.718120 \cdot 12}{1^2} \\ & \approx 7.837653307\end{aligned}$$
Modular invariants
Modular form 137904.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2695680 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $17$ | $1$ | $I_{3}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 204 = 2^{2} \cdot 3 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 199 & 6 \\ 198 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 101 & 0 \\ 0 & 203 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 20 & 87 \\ 1 & 52 \end{array}\right),\left(\begin{array}{rr} 65 & 198 \\ 93 & 185 \end{array}\right),\left(\begin{array}{rr} 107 & 198 \\ 108 & 197 \end{array}\right),\left(\begin{array}{rr} 105 & 2 \\ 10 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[204])$ is a degree-$22560768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/204\Z)$.
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 | $2$ | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 137904ba
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 17238o2, its twist by $-52$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.8619.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3788645211.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.107935903296.13 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.14263134912.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.140665008396638882505047105345681543359137822277632.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.273170519723525083135923218410222767046656.1 | \(\Z/6\Z\) | 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 | ord | ord | ord | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 2 | 2 | 4 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.