Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-231379008x-1468984984012\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-231379008xz^2-1468984984012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18741699675x-1070833828245750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(886530778/11881, 25794337956800/1295029)$ | $15.209698285248823874941245995$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 36400 \) | = | $2^{4} \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-139343750000000000000000000$ | = | $-1 \cdot 2^{19} \cdot 5^{24} \cdot 7^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{21405018343206000779641}{2177246093750000000} \) | = | $-1 \cdot 2^{-7} \cdot 5^{-18} \cdot 7^{-3} \cdot 13^{-1} \cdot 53^{3} \cdot 523877^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7564016255807022004709158945$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2585354888037067037533041064$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0145266852097035$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.622746142256209$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $15.209698285248823874941245995$ |
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| Real period: | $\Omega$ | ≈ | $0.019243935684060173582813006261$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2^{2}\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.0246669338069281910081901593 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.024666934 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.019244 \cdot 15.209698 \cdot 24}{1^2} \\ & \approx 7.024666934\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12192768 |
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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 not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 4 | 19 | 7 |
| $5$ | $2$ | $I_{18}^{*}$ | additive | 1 | 2 | 24 | 18 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|---|
| $3$ | 3B | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 19666 & 19665 \\ 29475 & 13096 \end{array}\right),\left(\begin{array}{rr} 13103 & 0 \\ 0 & 32759 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 32743 & 18 \\ 32742 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 24569 & 26190 \\ 0 & 32759 \end{array}\right),\left(\begin{array}{rr} 10306 & 19665 \\ 8415 & 13096 \end{array}\right),\left(\begin{array}{rr} 8201 & 22950 \\ 22770 & 23941 \end{array}\right),\left(\begin{array}{rr} 1516 & 19665 \\ 8055 & 19636 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$1051769626951680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 | $4$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | good | $2$ | \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 36400bs
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 910e1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.728.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.24676704000.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.228488000.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.1059968000.2 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.21320672256000000.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.305940066293694277300629238972416000000000.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.971460353775130433942454272000000000.1 | \(\Z/18\Z\) | 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 | ord | add | split | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 4 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.