Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-1629951265x-25327898503775\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-1629951265xz^2-25327898503775z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-132026052492x-18464434087409424\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-23345, 0)$ | $0$ | $2$ |
Integral points
\( \left(-23345, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 73920 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $2003069730816000000000000$ | = | $2^{27} \cdot 3^{8} \cdot 5^{12} \cdot 7 \cdot 11^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{1826870018430810435423307849}{7641104625000000000} \) | = | $2^{-9} \cdot 3^{-8} \cdot 5^{-12} \cdot 7^{-1} \cdot 11^{-3} \cdot 109^{3} \cdot 601^{3} \cdot 18661^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8719617908848116897483424946$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8322410200448937256224943124$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0452427382284861$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.712229484804686$ | |||
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: | $\Omega$ | ≈ | $0.023760523711824156315268820982$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7107577072513392546993551107 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.710757707 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.023761 \cdot 1.000000 \cdot 288}{2^2} \\ & \approx 1.710757707\end{aligned}$$
Modular invariants
Modular form 73920.2.a.dx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 31850496 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{17}^{*}$ | additive | 1 | 6 | 27 | 9 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $2$ | 2B | 4.12.0.8 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 3697 & 24 \\ 7404 & 289 \end{array}\right),\left(\begin{array}{rr} 3376 & 21 \\ 2235 & 8866 \end{array}\right),\left(\begin{array}{rr} 2311 & 24 \\ 2322 & 289 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1336 & 21 \\ 1035 & 8866 \end{array}\right),\left(\begin{array}{rr} 7314 & 8831 \\ 5005 & 6544 \end{array}\right),\left(\begin{array}{rr} 7701 & 6164 \\ 20 & 4701 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 77 = 7 \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 448 = 2^{6} \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 14784 = 2^{6} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 10560 = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 73920bt
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310g7, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{154}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-154}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{154})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.934979584.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{154})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-154})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.72589644288.26 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.186606965293056.195 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.366189066569178420539686121743823241216000000000000.2 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | add | nonsplit | split | split | split |
| $\lambda$-invariant(s) | - | 6 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.