Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+752237x-113781983\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+752237xz^2-113781983z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+974899125x-5311536896250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(558, 21625\right) \) | $1.1609423611467510997128045175$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([558:21625:1]\) | $1.1609423611467510997128045175$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(20091, 4731264\right) \) | $1.1609423611467510997128045175$ | $\infty$ |
Integral points
\( \left(366, 14329\right) \), \( \left(366, -14695\right) \), \( \left(558, 21625\right) \), \( \left(558, -22183\right) \)
\([366:14329:1]\), \([366:-14695:1]\), \([558:21625:1]\), \([558:-22183:1]\)
\((13179,\pm 3134592)\), \((20091,\pm 4731264)\)
Invariants
| Conductor: | $N$ | = | \( 68450 \) | = | $2 \cdot 5^{2} \cdot 37^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-32841298035200000000$ | = | $-1 \cdot 2^{15} \cdot 5^{8} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{46969655}{32768} \) | = | $2^{-15} \cdot 5 \cdot 211^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4333844607293357224282247406$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44503310388217674948966265040$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0629647337743247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.688942430648762$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1609423611467510997128045175$ |
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| Real period: | $\Omega$ | ≈ | $0.11724360072845592508422726653$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 90 $ = $ ( 3 \cdot 5 )\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.250175639313651200362062224 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.250175639 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.117244 \cdot 1.160942 \cdot 90}{1^2} \\ & \approx 12.250175639\end{aligned}$$
Modular invariants
Modular form 68450.2.a.bm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1555200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $37$ | $2$ | $I_0^{*}$ | 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.4.1 | 5.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1481 & 2960 \\ 1480 & 2961 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1110 & 1 \end{array}\right),\left(\begin{array}{rr} 1481 & 3330 \\ 1480 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2220 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3960 & 1 \end{array}\right),\left(\begin{array}{rr} 481 & 3960 \\ 480 & 481 \end{array}\right),\left(\begin{array}{rr} 1 & 3996 \\ 2220 & 1 \end{array}\right),\left(\begin{array}{rr} 1479 & 2590 \\ 2960 & 2663 \end{array}\right),\left(\begin{array}{rr} 3886 & 1665 \\ 3885 & 1 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 4439 \end{array}\right),\left(\begin{array}{rr} 1 & 3552 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2442 \\ 3330 & 2221 \end{array}\right),\left(\begin{array}{rr} 2221 & 3330 \\ 3885 & 3331 \end{array}\right),\left(\begin{array}{rr} 1111 & 3330 \\ 2775 & 3331 \end{array}\right)$.
The torsion field $K:=\Q(E[4440])$ is a degree-$167931740160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4440\Z)$.
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 | $4$ | \( 34225 = 5^{2} \cdot 37^{2} \) |
| $3$ | good | $2$ | \( 34225 = 5^{2} \cdot 37^{2} \) |
| $5$ | additive | $14$ | \( 1369 = 37^{2} \) |
| $37$ | additive | $686$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 68450.bm
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50.b4, its twist by $185$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-111}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{185}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-111})\) | \(\Z/15\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.23078773125.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.54705240000.5 | \(\Z/6\Z\) | not in database |
| $6$ | 6.2.10130600000.1 | \(\Z/10\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/30\Z\) | not in database |
| $18$ | 18.0.991035916125874083964008999000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.3222389810092767432489792000000000000.1 | \(\Z/6\Z\) | not in database |
| $20$ | 20.0.22391715889879730530083179473876953125.1 | \(\Z/5\Z\) | 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 | add | ord | ord | ord | ord | ord | ord | ss | ord | add | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 5 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | - | 1 | 1 | 1 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.