Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-54421x+4945517\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-54421xz^2+4945517z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-70530291x+231795992142\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 338 \) | = | $2 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-347488235454464$ | = | $-1 \cdot 2^{15} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{1680914269}{32768} \) | = | $-1 \cdot 2^{-15} \cdot 29^{3} \cdot 41^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5836155373581373101277390737$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.34009648073801524191237650747$ |
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| $abc$ quality: | $Q$ | ≈ | $1.023217716971265$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.618054134929045$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.53962733913408357725916763796$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0792546782681671545183352759 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.079254678 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.539627 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 1.079254678\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1560 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{15}$ | nonsplit multiplicative | 1 | 1 | 15 | 15 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
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 |
|---|---|---|
| $3$ | 3Ns | 3.6.0.1 |
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 16 & 375 \\ 375 & 1306 \end{array}\right),\left(\begin{array}{rr} 781 & 0 \\ 0 & 781 \end{array}\right),\left(\begin{array}{rr} 481 & 480 \\ 1080 & 481 \end{array}\right),\left(\begin{array}{rr} 1351 & 630 \\ 0 & 1351 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 1511 & 855 \\ 1365 & 1556 \end{array}\right),\left(\begin{array}{rr} 781 & 1248 \\ 300 & 781 \end{array}\right),\left(\begin{array}{rr} 937 & 150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 781 & 150 \\ 795 & 691 \end{array}\right),\left(\begin{array}{rr} 657 & 1367 \\ 250 & 783 \end{array}\right),\left(\begin{array}{rr} 521 & 150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 0 \\ 0 & 241 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 361 & 1410 \\ 750 & 181 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1080 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$1610219520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 13 \) |
| $3$ | good | $2$ | \( 169 = 13^{2} \) |
| $5$ | good | $2$ | \( 169 = 13^{2} \) |
| $13$ | additive | $62$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 338.b
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 338.d1, its twist by $13$.
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 |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.0.19773.1 | \(\Z/3\Z\) | not in database |
| $4$ | 4.2.6591.1 | \(\Z/3\Z\) | not in database |
| $4$ | 4.4.274625.2 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.0.390971529.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.2.1423311812421484544.43 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.0.4881467152985645008087158203125.1 | \(\Z/5\Z\) | 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 | nonsplit | ord | ord | ord | ss | add | ord | ord | ord | ss | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | 0 | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0,0 | 2 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.