Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1644x+24972\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1644xz^2+24972z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2130651x+1197061686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(279, 4484\right) \) | $4.1733594099660855352851218414$ | $\infty$ |
| \( \left(24, -4\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([279:4484:1]\) | $4.1733594099660855352851218414$ | $\infty$ |
| \([24:-4:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10059, 998784\right) \) | $4.1733594099660855352851218414$ | $\infty$ |
| \( \left(879, 1836\right) \) | $0$ | $4$ |
Integral points
\( \left(23, -12\right) \), \( \left(24, -4\right) \), \( \left(24, -21\right) \), \( \left(279, 4484\right) \), \( \left(279, -4764\right) \)
\([23:-12:1]\), \([24:-4:1]\), \([24:-21:1]\), \([279:4484:1]\), \([279:-4764:1]\)
\( \left(843, 0\right) \), \((879,\pm 1836)\), \((10059,\pm 998784)\)
Invariants
| Conductor: | $N$ | = | \( 4641 \) | = | $3 \cdot 7 \cdot 13 \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $78897$ | = | $3 \cdot 7 \cdot 13 \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{491411892194497}{78897} \) | = | $3^{-1} \cdot 7^{-1} \cdot 13^{-1} \cdot 17^{-2} \cdot 23^{3} \cdot 47^{3} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.34216591193674093748940248816$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.34216591193674093748940248816$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9062912298209369$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.006818117512229$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.1733594099660855352851218414$ |
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| Real period: | $\Omega$ | ≈ | $2.6911365006591067772943318172$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.4038849798161107521651937587 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.403884980 \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 2.691137 \cdot 4.173359 \cdot 2}{4^2} \\ & \approx 1.403884980\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1536 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2B | 8.24.0.45 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 74256 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 34957 & 16 \\ 8480 & 73941 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 18572 & 18693 \end{array}\right),\left(\begin{array}{rr} 68552 & 1 \\ 45775 & 10 \end{array}\right),\left(\begin{array}{rr} 74241 & 16 \\ 74240 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 74158 & 74243 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 63664 & 5 \\ 52995 & 74242 \end{array}\right),\left(\begin{array}{rr} 27857 & 16 \\ 9218 & 55599 \end{array}\right),\left(\begin{array}{rr} 49520 & 5 \\ 74211 & 74242 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 74252 & 74253 \end{array}\right)$.
The torsion field $K:=\Q(E[74256])$ is a degree-$25429452313853952$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/74256\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 273 = 3 \cdot 7 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1547 = 7 \cdot 13 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 663 = 3 \cdot 13 \cdot 17 \) |
| $13$ | split multiplicative | $14$ | \( 357 = 3 \cdot 7 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 273 = 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 4641.b
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{357}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{221}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{221}, \sqrt{273})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | ord | nonsplit | ord | nonsplit | ord | split | split | ord | ss | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.