Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-36275x-2674475\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-36275xz^2-2674475z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-47013075x-124075112850\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 14450 \) | = | $2 \cdot 5^{2} \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-120687845000$ | = | $-1 \cdot 2^{3} \cdot 5^{4} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{349938025}{8} \) | = | $-1 \cdot 2^{-3} \cdot 5^{2} \cdot 241^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2398132202182813530685645474$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.71327275595452681192312253928$ |
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| $abc$ quality: | $Q$ | ≈ | $1.050775872125697$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.500763612283086$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.17296791452473675078805590014$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.51890374357421025236416770043 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.518903744 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.172968 \cdot 1.000000 \cdot 3}{1^2} \\ & \approx 0.518903744\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 30240 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $17$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $5$ | 5B.4.2 | 5.12.0.2 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1361 & 680 \\ 1360 & 681 \end{array}\right),\left(\begin{array}{rr} 1021 & 1530 \\ 765 & 1531 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1020 & 1 \end{array}\right),\left(\begin{array}{rr} 1921 & 120 \\ 1920 & 1921 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 120 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1836 \\ 1020 & 1 \end{array}\right),\left(\begin{array}{rr} 1799 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 1361 & 1530 \\ 1360 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1632 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 510 & 1 \end{array}\right),\left(\begin{array}{rr} 1633 & 1530 \\ 0 & 409 \end{array}\right),\left(\begin{array}{rr} 766 & 1785 \\ 765 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1122 \\ 1530 & 1021 \end{array}\right),\left(\begin{array}{rr} 511 & 1530 \\ 255 & 1531 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$7219445760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $4$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $3$ | good | $2$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 578 = 2 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 14450l
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50a2, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-51}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-170 +34 \sqrt{5}})\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.2238485625.5 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.5306040000.15 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.105706265625.1 | \(\Z/15\Z\) | not in database |
| $10$ | 10.2.17748212500000000.1 | \(\Z/5\Z\) | 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 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.904303412765472167650539000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.2940375878524118210112000000000000.1 | \(\Z/6\Z\) | not in database |
| $20$ | 20.0.18600378723064531406250000000000000000.1 | \(\Z/15\Z\) | 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 | add | ord | ord | ord | add | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 0 | - | 0 | 0 | 0 | - | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 1 | - | 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 $0$.