Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-322x+2127\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-322xz^2+2127z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-417339x+105505686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 23)$ | $0.030736550181821687635348975624$ | $\infty$ |
Integral points
\( \left(-21, 23\right) \), \( \left(-21, -3\right) \), \( \left(-7, 67\right) \), \( \left(-7, -61\right) \), \( \left(5, 23\right) \), \( \left(5, -29\right) \), \( \left(9, 3\right) \), \( \left(9, -13\right) \), \( \left(15, 23\right) \), \( \left(15, -39\right) \), \( \left(17, 35\right) \), \( \left(17, -53\right) \), \( \left(57, 387\right) \), \( \left(57, -445\right) \), \( \left(1373, 50207\right) \), \( \left(1373, -51581\right) \)
Invariants
| Conductor: | $N$ | = | \( 338 \) | = | $2 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-71991296$ | = | $-1 \cdot 2^{15} \cdot 13^{3} $ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.30114085862736894210099535293$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.34009648073801524191237650746$ |
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| $abc$ quality: | $Q$ | ≈ | $1.023217716971265$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.975159604987742$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.030736550181821687635348975624$ |
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| Real period: | $\Omega$ | ≈ | $1.9456540408901337455759826091$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 30 $ = $ ( 3 \cdot 5 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7940807919285122494029160801 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.794080792 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.945654 \cdot 0.030737 \cdot 30}{1^2} \\ & \approx 1.794080792\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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
$\left(\begin{array}{rr} 16 & 375 \\ 375 & 1306 \end{array}\right),\left(\begin{array}{rr} 781 & 0 \\ 0 & 781 \end{array}\right),\left(\begin{array}{rr} 481 & 480 \\ 1080 & 481 \end{array}\right),\left(\begin{array}{rr} 1351 & 630 \\ 0 & 1351 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 1511 & 855 \\ 1365 & 1556 \end{array}\right),\left(\begin{array}{rr} 781 & 1248 \\ 300 & 781 \end{array}\right),\left(\begin{array}{rr} 937 & 150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 903 & 193 \\ 1310 & 777 \end{array}\right),\left(\begin{array}{rr} 781 & 150 \\ 795 & 691 \end{array}\right),\left(\begin{array}{rr} 521 & 150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 0 \\ 0 & 241 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 361 & 1410 \\ 750 & 181 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1080 & 1 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 13 \) |
| $3$ | good | $2$ | \( 169 = 13^{2} \) |
| $5$ | good | $2$ | \( 169 = 13^{2} \) |
| $13$ | additive | $50$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 338e
consists of 2 curves linked by isogenies of
degree 5.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
| $4$ | 4.4.274625.1 | \(\Z/5\Z\) | 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 |
| $12$ | 12.2.1423311812421484544.1 | \(\Z/4\Z\) | 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 | split | ord | ord | ord | ss | add | ord | ord | ord | ss | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1 | 1,1 | - | 1 | 1 | 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 | 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.