Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-23675x-1401750\)
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(homogenize, simplify) |
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\(y^2z=x^3-23675xz^2-1401750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-23675x-1401750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{351}{4}, \frac{57}{8}\right) \) | $4.6009492369241822740352406890$ | $\infty$ |
| \( \left(-90, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-702:57:8]\) | $4.6009492369241822740352406890$ | $\infty$ |
| \([-90:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{351}{4}, \frac{57}{8}\right) \) | $4.6009492369241822740352406890$ | $\infty$ |
| \( \left(-90, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-90, 0\right) \)
\([-90:0:1]\)
\( \left(-90, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 4400 \) | = | $2^{4} \cdot 5^{2} \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $440000000000$ | = | $2^{12} \cdot 5^{10} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( \frac{22930509321}{6875} \) | = | $3^{3} \cdot 5^{-4} \cdot 11^{-1} \cdot 947^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2122006568022365960546243784$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.28566547997475890066298740967$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0771710471613196$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.986092942113602$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6009492369241822740352406890$ |
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| Real period: | $\Omega$ | ≈ | $0.38488844392434208704704456488$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5417043847492752891152577495 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.541704385 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.384888 \cdot 4.600949 \cdot 8}{2^2} \\ & \approx 3.541704385\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 8.12.0.6 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 440 = 2^{3} \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 263 & 432 \\ 172 & 407 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 433 & 8 \\ 432 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 404 & 1 \\ 343 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 434 & 435 \end{array}\right),\left(\begin{array}{rr} 272 & 47 \\ 247 & 200 \end{array}\right),\left(\begin{array}{rr} 59 & 58 \\ 178 & 395 \end{array}\right)$.
The torsion field $K:=\Q(E[440])$ is a degree-$202752000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/440\Z)$.
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 | $2$ | \( 275 = 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 176 = 2^{4} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 4400q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 55a3, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | 2.0.20.1-605.1-b4 |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.2129600.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.72563138560000.21 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.309760000.3 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.72563138560000.35 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.128079468000000.1 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.1404822362521600000000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ss | add | ss | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 5,1 | - | 1,1 | 2 | 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 |
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.